Biomolecular Feedback Systems

Transcription

1 Biomolecular Feeback Systems Domitilla Del Vecchio MIT Richar M. Murray Caltech Version 1.0b, September 14, 2014 c 2014 by Princeton University Press All rights reserve. This is the electronic eition of Biomolecular Feeback Systems, availablefrom murray/bfswiki. Printe versions are available from Princeton University Press, This manuscript is for personal use only an may not be reprouce, in whole or in part, without written consent from the publisher (see

2 Chapter 2 Dynamic Moeling of Core Processes The goal of this chapter is to escribe basic biological mechanisms in a way that can be represente by simple ynamical moels. We begin the chapter with a iscussion of the basic moeling formalisms that we will utilize to moel biomolecular feeback systems. We then procee to stuy a number of core processes within the cell, proviing ifferent moel-base escriptions of the ynamics that will be use in later chapters to analyze an esign biomolecular systems. The focus in this chapter an the next is on eterministic moels using orinary ifferential equations; Chapter 4 escribes how to moel the stochastic nature of biomolecular systems. 2.1 Moeling chemical reactions In orer to evelop moels for some of the core processes of the cell, we will nee to buil up a basic escription of the biochemical reactions that take place, incluing prouction an egraation of proteins, regulation of transcription an translation, an intracellular sensing, action an computation. As in other isciplines, biomolecular systems can be moele in a variety of ifferent ways, at many ifferent levels of resolution, as illustrate in Figure 2.1. The choice of which moel to use epens on the questions that we want to answer, an goo moeling takes practice, experience, an iteration. We must properly capture the aspects of the system that are important, reason about the appropriate temporal an spatial scales to be inclue, an take into account the types of simulation an analysis tools to be applie. Moels use for analyzing existing systems shoul make testable preictions an provie insight into the unerlying ynamics. Design moels must aitionally capture enough of the important behavior to allow ecisions regaring how to interconnect subsystems, choose parameters an esign regulatory elements. In this section we escribe some of the basic moeling frameworks that we will buil on throughout the rest of the text. We begin with brief escriptions of the relevant physics an chemistry of the system, an then quickly move to moels that focus on capturing the behavior using reaction rate equations. In this chapter our emphasis will be on ynamics with time scales measure in secons to hours an mean behavior average across a large number of molecules. We touch only briefly on moeling in the case where stochastic behavior ominates an efer a more etaile treatment until Chapter 4.

3 30 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Granularity Molecular counts Concentration Fast Chemical Master Equation Fokker-Planck Equation Chemical Langevin Equation Timescale Reaction Rate Equations Statistical Mechanics Slow Figure 2.1: Different methos of moeling biomolecular systems. Reaction kinetics At the fine en of the moeling scale, we can attempt to moel the molecular ynamics of the cell, in which we attempt to moel the iniviual proteins an other species an their interactions via molecular-scale forces an motions. At this scale, the iniviual interactions between protein omains, DNA an RNA are resolve, resulting in a highly etaile moel of the ynamics of the cell. For our purposes in this text, we will not require the use of such a etaile scale an we will consier the main moeling formalisms epicte in Figure 2.1. We start with the abstraction of molecules that interact with each other through stochastic events that are guie by the laws of thermoynamics. We begin with an equilibrium point of view, commonly referre to as statistical mechanics, an then briefly escribe how to moel the (statistical) ynamics of the system using chemical kinetics. We cover both of these points of view very briefly here, primarily as a stepping stone to eterministic moels. The unerlying representation for both statistical mechanics an chemical kinetics is to ientify the appropriate microstates of the system. A microstate correspons to a given configuration of the components (species) in the system relative to each other an we must enumerate all possible configurations between the molecules that are being moele. As an example, consier the istribution of RNA polymerase in the cell. It is known that most RNA polymerases are boun to the DNA in a cell, either as they prouce RNA or as they iffuse along the DNA in search of a promoter site. Hence we can moel the microstates of the RNA polymerase system as all possible locations of the RNA polymerase in the cell, with the vast majority of these corresponing to the RNA polymerase at some location on the DNA. This is illustrate

4 2.1. MODELING CHEMICAL REACTIONS 31 RNA polymerase DNA Microstate 1 Microstate 2 Microstate 3 etc. Figure 2.2: Microstates for RNA polymerase. Each microstate of the system correspons to the RNA polymerase being locate at some position in the cell. If we iscretize the possible locations on the DNA an in the cell, the microstate correspons to all possible non-overlapping locations of the RNA polymerases. Figure aapte from Phillips, Konev an Theriot [78]. in Figure 2.2. In statistical mechanics, we moel the configuration of the cell by the probability that the system is in a given microstate. This probability can be calculate base on the energy levels of the ifferent microstates. The laws of statistical mechanics state that if we have a set of microstates Q, then the steay state probability that the system is in a particular microstate q is given by P(q) = 1 Z e E q/(k B T), (2.1) where E q is the energy associate with the microstate q Q, k B is the Boltzmann constant, T is the temperature in egrees Kelvin, an Z is a normalizing factor, known as the partition function, Z = e E q/(k B T). q Q By keeping track of those microstates that correspon to a given system state (also calle a macrostate), we can compute the overall probability that a given macrostate is reache. Thus, if we have a set of states S Q that correspons to a given macrostate, then the probability of being in the set S is given by P(S ) = 1 e E q/(k B T) = Z q Q e E q/(k B T). (2.2) q S q S e E q/(k B T)

5 32 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES This can be use, for example, to compute the probability that some RNA polymerase is boun to a given promoter, average over many inepenent samples, an from this we can reason about the rate of expression of the corresponing gene. Statistical mechanics escribes the steay state istribution of microstates, but oes not tell us how the microstates evolve in time. To inclue the ynamics, we must consier the chemical kinetics of the system an moel the probability that we transition from one microstate to another in a given perio of time. Let q represent the microstate of the system, which we shall take as a vector of integers that represents the number of molecules of a specific type (species) in given configurations or locations. Assume we have a set of m chemical reactions Rj, j = 1,...,M, in which a chemical reaction is a process that leas to the transformation of one set of chemical species to another one. We use ξ j to represent the change in state q associate with reaction Rj. We escribe the kinetics of the system by making use of the propensity function a j (q,t) associate with reaction Rj, which captures the instantaneous probability that at time t a system will transition between state q an state q + ξ j. More specifically, the propensity function is efine such that a j (q,t) = Probability that reaction Rj will occur between time t an time t + given that the microstate is q. We will give more etail in Chapter 4 regaring the valiity of this functional form, but for now we simply assume that such a function can be efine for our system. Using the propensity function, we can keep track of the probability istribution for the state by looking at all possible transitions into an out of the current state. Specifically, given P(q,t), the probability of being in state q at time t,wecan compute the time erivative P(q,t)/ as P M (q,t) = ( a j (q ξ j,t)p(q ξ j,t) a j (q,t)p(q,t) ). (2.3) j=1 This equation (an its variants) is calle the chemical master equation (CME). The first sum on the right-han sie represents the transitions into the state q from some other state q ξ j an the secon sum represents the transitions out of the state q. The ynamics of the istribution P(q,t) epen on the form of the propensity functions a j (q,t). Consier a simple reversible reaction of the form A + B AB, (2.4) in which a molecule of A an a molecule of B come together to form the complex AB, in which A an B are boun to each other, an this complex can, in turn, issociate back into the A an B species. In the sequel, to make notation easier,

6 2.1. MODELING CHEMICAL REACTIONS 33 we will sometimes represent the complex AB as A : B. It is often useful to write reversible reactions by splitting the forwar reaction from the backwar reaction: Rf: A+ B AB, Rr: AB A + B. (2.5) We assume that the reaction takes place in a well-stirre volume Ω an let the configurations q be represente by the number of each species that is present. The forwar reaction Rf is a bimolecular reaction an we will see in Chapter 4 that it has a propensity function a f (q) = k f Ω n An B, where k f is a parameter that epens on the forwar reaction, an n A an n B are the number of molecules of each species. The reverse reaction Rr is a unimolecular reaction an we will see that it has a propensity function a r (q) = k r n AB, where k r is a parameter that epens on the reverse reaction an n AB is the number of molecules of AB that are present. If we now let q = (n A,n B,n AB ) represent the microstate of the system, then we can write the chemical master equation as P (n A,n B,n AB ) = k r (n AB + 1)P(n A 1,n B 1,n AB + 1) k f Ω n An B P(n A,n B,n AB ) + k f Ω (n A + 1)(n B + 1)P(n A + 1,n B + 1,n AB 1) k r n AB P(n A,n B,n AB ). The first an thir terms on the right-han sie represent the transitions into the microstate q = (n A,n B,n AB ) an the secon an fourth terms represent the transitions out of that state. The number of ifferential equations epens on the number of molecules of A, B an AB that are present. For example, if we start with one molecule of A, one molecule of B, an three molecules of AB, then the possible states an ynamics are q 0 = (1,0,4), P 0 / = 2(k f /Ω)P 1 4k r P 0 q 1 = (2,1,3), P 1 / = 4k r P 0 2(k f /Ω)P 1 + 6(k f /Ω)P 2 3k r P 1, q 2 = (3,2,2), P 2 / = 3k r P 1 6(k f /Ω)P 2,+12(k f /Ω)P 3 2k r P 2, q 3 = (4,3,1), P 3 / = 2k r P 2 12(k f /Ω)P 3,+20(k f /Ω)P 4 1k r P 3, q 4 = (5,4,0), P 4 / = 1k r P 3 20(k f /Ω)P 4,

7 34 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES where P i = P(q i,t). Note that the states of the chemical master equation are the probabilities that we are in a specific microstate, an the chemical master equation is a linear ifferential equation (we see from equation (2.3) that this is true in general). The primary ifference between the statistical mechanics escription given by equation (2.1) an the chemical kinetics escription in equation (2.3) is that the master equation formulation escribes how the probability of being in a given microstate evolves over time. Of course, if the propensity functions an energy levels are moele properly, the steay state, average probabilities of being in a given microstate, shoul be the same for both formulations. Reaction rate equations Although very general in form, the chemical master equation suffers from being a very high-imensional representation of the ynamics of the system. We shall see in Chapter 4 how to implement simulations that obey the master equation, but in many instances we will not nee this level of etail in our moeling. In particular, there are many situations in which the number of molecules of a given species is such that we can reason about the behavior of a chemically reacting system by keeping track of the concentration of each species as a real number. This is of course an approximation, but if the number of molecules is sufficiently large, then the approximation will generally be vali an our moels can be ramatically simplifie. To go from the chemical master equation to a simplifie form of the ynamics, we begin by making a number of assumptions. First, we assume that we can represent the state of a given species by its concentration n A /Ω, where n A is the number of molecules of A in a given volume Ω. We also treat this concentration as a real number, ignoring the fact that the real concentration is quantize. Finally, we assume that our reactions take place in a well-stirre volume, so that the rate of interactions between two species is solely etermine by the concentrations of the species. Before proceeing, we shoul recall that in many (an perhaps most) situations insie of cells, these assumptions are not particularly goo ones. Biomolecular systems often have very small molecular counts an are anything but well mixe. Hence, we shoul not expect that moels base on these assumptions shoul perform well at all. However, experience inicates that in many cases the basic form of the equations provies a goo moel for the unerlying ynamics an hence we often fin it convenient to procee in this manner. Putting asie our potential concerns, we can now create a moel for the ynamics of a system consisting of a set of species S i, i = 1,...,n, unergoing a set of reactions Rj, j = 1,...,M. We write x i = [S i ] = n Si /Ω for the concentration of species i (viewe as a real number). Because we are intereste in the case where

8 2.1. MODELING CHEMICAL REACTIONS 35 the number of molecules is large, we no longer attempt to keep track of every possible configuration, but rather simply assume that the state of the system at any given time is given by the concentrations x i. Hence the state space for our system is given by x R n an we seek to write our ynamics in the form of an orinary ifferential equation (ODE) x = f (x,θ), where θ R p represents the vector of parameters that govern ynamic behavior an f : R n R p R n escribes the rate of change of the concentrations as a function of the instantaneous concentrations an parameter values. To illustrate the general form of the ynamics, we consier again the case of a basic bimolecular reaction A + B AB. Each time the forwar reaction occurs, we ecrease the number of molecules of A an B by one an increase the number of molecules of AB (a separate species) by one. Similarly, each time the reverse reaction occurs, we ecrease the number of molecules of AB by one an increase the number of molecules of A an B. Using our iscussion of the chemical master equation, we know that the likelihoo that the forwar reaction occurs in a given interval is given by a f (q) = (k f /Ω)n A n B an the reverse reaction has likelihoo a r (q) = k r n AB. If we assume that n AB is a real number instea of an integer an ignore some of the formalities of ranom variables, we can escribe the evolution of n AB using the equation n AB (t + ) = n AB (t) + a f (q) a r (q). Here we let q be the state of the system with the number of molecules of AB equal to n AB. Roughly speaking, this equation states that the (approximate) number of molecules of AB at time t + compare with time t increases by the probability that the forwar reaction occurs in time an ecreases by the probability that the reverse reaction occurs in that perio. To convert this expression into an equivalent one for the concentration of the species AB, we write [AB] = n AB /Ω, [A]= n A /Ω, [B]= n B /Ω, an substitute the expressions for a f (q) an a r (q): [AB](t + ) [AB](t) = ( a f (q,t) a r (q) ) /Ω Taking the limit as approaches zero, we obtain = ( k f n A n B /Ω 2 k r n AB /Ω ) = ( k f [A][B] k r [AB] ). [AB] = k f[a][b] k r [AB].

9 36 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Our erivation here has skippe many important steps, incluing a careful erivation using ranom variables an some assumptions regaring the way in which approaches zero. These are escribe in more etail when we erive the chemical Langevin equation (CLE) in Chapter 4, but the basic form of the equations are correct uner the assumptions that the reactions are well-stirre an the molecular counts are sufficiently large. In a similar fashion we can write equations to escribe the ynamics of A an B an the entire system of equations is given by [A] = k r [AB] k f [A][B], [B] = k r [AB] k f [A][B], [AB] = k f [A][B] k r [AB], or A = k rc k f A B, B = k rc k f A B, C = k fa B k r C, where C = [AB], A = [A], an B = [B]. These equations are known as the mass action kinetics or the reaction rate equations for the system. The parameters k f an k r are calle the rate constants an they match the parameters that were use in the unerlying propensity functions. Note that the same rate constants appear in each term, since the rate of prouction of AB must match the rate of epletion of A an B an vice versa. We aopt the stanar notation for chemical reactions with specifie rates an write the iniviual reactions as A + B k f AB, AB k r A + B, where k f an k r are the reaction rate constants. For biirectional reactions we can also write A + B k f AB. It is easy to generalize these ynamics to more complex reactions. For example, if we have a reversible reaction of the form kr A + 2B k f 2C+ D, kr where A, B, C an D are appropriate species an complexes, then the ynamics for the species concentrations can be written as A = k rc 2 D k f A B 2, B = 2k rc 2 D 2k f A B 2, C = 2k fa B 2 2k r C 2 D, D = k fa B 2 k r C 2 D. (2.6)

10 2.1. MODELING CHEMICAL REACTIONS 37 Rearranging this equation, we can write the ynamics as A 1 1 B 2 2 = C 2 2 k fa B 2 k r C 2 D. (2.7) D 1 1 We see that in this ecomposition, the first term on the right-han sie is a matrix of integers reflecting the stoichiometry of the reactions an the secon term is a vector of rates of the iniviual reactions. More generally, given a chemical reaction consisting of a set of species S i, i = 1,...,n an a set of reactions Rj, j = 1,...,M, we can write the mass action kinetics in the form x = Nv(x), where N R n M is the stoichiometry matrix for the system an v(x) R M is the reaction flux vector. Each row of v(x) correspons to the rate at which a given reaction occurs an the corresponing column of the stoichiometry matrix correspons to the changes in concentration of the relevant species. For example, for the system in equation (2.7) we have x = (A, B,C, D), N =, v(x) = 2 2 k fa B 2 k r C D. The conservation of species is at the basis of reaction rate moels since species are usually transforme, but are not create from nothing or estroye. Even the basic process of protein egraation transforms a protein of interest A into a prouct X that is not use in any other reaction. Specifically, the egraation rate of a protein is etermine by the amounts of proteases present, which bin to recognition sites (egraation tags) an then egrae the protein. Degraation of a protein A by a protease P can then be moele by the following two-step reaction: A + P a AP k P + X. As a result of the reaction, protein A has isappeare, so that this reaction is often simplifie to A. Similarly, the birth of a molecule is a complicate process that involves many reactions an species, as we will see later in this chapter. When the process that creates a species of interest A is not relevant for the problem uner stuy, we will use the shorter escription of a birth reaction given by k f A

11 38 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES ATP bining Kinase ADP release Kinase ATP Bining + catalysis Kinase Substrate ADP P Substrate Substrate release Substrate P Figure 2.3: Phosphorylation of a protein via a kinase. In the process of phosphorylation, aproteincalleakinasebinstoatp(aenosinetriphosphate) an transfers one of the phosphate groups (P) from ATP to a substrate, hence proucing aphosphorylate substrate an ADP (aenosine iphosphate). Figure aapte from Mahani [63]. an escribe its ynamics using the ifferential equation A = k f. Example 2.1 (Covalent moification of a protein). Consier the set of reactions involve in the phosphorylation of a protein by a kinase, as shown in Figure 2.3. Let S represent the substrate, K represent the kinase an S * represent the phosphorylate (activate) substrate. The sets of reactions illustrate in Figure 2.3 are R1: K+ ATP K:ATP, R2: K:ATP K + ATP, R3: S+ K:ATP S:K:ATP, R4: S:K:ATP S + K:ATP, R5: S:K:ATP S :K:ADP, R6: S :K:ADP S + K:ADP, R7: K:ADP K + ADP, R8: K+ ADP K:ADP. We now write the kinetics for each reaction: v 1 = k 1 [K][ATP], v 2 = k 2 [K:ATP], v 3 = k 3 [S][K:ATP], v 4 = k 4 [S:K:ATP], v 5 = k 5 [S:K:ATP], v 6 = k 6 [S :K:ADP], v 7 = k 7 [K:ADP], v 8 = k 8 [K][ADP]. We treat [ATP] as a constant (regulate by the cell) an hence o not irectly track its concentration. (If esire, we coul similarly ignore the concentration of

12 2.1. MODELING CHEMICAL REACTIONS 39 ADP since we have chosen not to inclue the many aitional reactions in which it participates.) The kinetics for each species are thus given by [K] = v 1 + v 2 + v 7 v 8, [K:ATP] = v 1 v 2 v 3 + v 4, [S] = v 3 + v 4, [S:K:ATP] = v 3 v 4 v 5, [S ] = v 6, [S :K:ADP] = v 5 v 6, [ADP] = v 7 v 8, [K:ADP] = v 6 v 7 + v 8. Collecting these equations together an writing the state as a vector, we obtain [K] [K:ATP] [S] [S:K:ATP] [S ] [S :K:ADP] [ADP] [K:ADP] } {{ } x = } {{ } N v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8, }{{} v(x) which is in stanar stoichiometric form. Reuce-orer mechanisms In this section, we look at the ynamics of some common reactions that occur in biomolecular systems. Uner some assumptions on the relative rates of reactions an concentrations of species, it is possible to erive reuce-orer expressions for the ynamics of the system. We focus here on an informal erivation of the relevant results, but return to these examples in the next chapter to illustrate that the same results can be erive using a more formal an rigorous approach. Simple bining reaction. Consier the reaction in which two species A an B bin reversibly to form a complex C = AB: A + B a C, (2.8) where a is the association rate constant an is the issociation rate constant. Assume that B is a species that is controlle by other reactions in the cell an that the total concentration of A is conserve, so that A + C = [A] + [AB] = A tot. If the

13 40 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES ynamics of this reaction are fast compare to other reactions in the cell, then the amount of A an C present can be compute as a (steay state) function of the amount of B. To compute how A an C epen on the concentration of B at the steay state, we must solve for the equilibrium concentrations of A an C. The rate equation for C is given by C = ab A C = ab (A tot C) C. By setting C/ = 0 an letting K := /a, we obtain the expressions C = A tot(b/k ) 1 + (B/K ), A = A tot 1 + (B/K ). The constant K is calle the issociation constant of the reaction. Its inverse measures the affinity of A bining to B. The steay state value of C increases with B while the steay state value of A ecreases with B as more of A is foun in the complex C. Note that when B K, A an C have equal concentration. Thus the higher the value of K, the more B is require for A to form the complex C. K has the units of concentration an it can be interprete as the concentration of B at which half of the total number of molecules of A are associate with B. Therefore a high K represents a weak affinity between A an B, while a low K represents a strong affinity. Cooperative bining reaction. Assume now that B bins to A only after imerization, that is, only after bining another molecule of B. Then, we have that reactions (2.8) become B + B k 1 B 2, B 2 + A a C, A +C = A tot, k2 in which B 2 = B : B represents the imer of B, that is, the complex of two molecules of B boun to each other. The corresponing ODE moel is given by B 2 = k 1 B 2 k 2 B 2 ab 2 (A tot C) + C, C = ab 2 (A tot C) C. By setting B 2 / = 0, C/ = 0, an by efining K m := k 2 /k 1, we obtain that B 2 = B 2 /K m, C = A tot(b 2 /K ) 1 + (B 2 /K ), A = A tot 1 + (B 2 /K ), so that C = A totb 2 /(K m K ) 1 + B 2 /(K m K ), A = A tot 1 + B 2 /(K m K ).

14 2.1. MODELING CHEMICAL REACTIONS 41 C A n = 1 n = 5 n = Normalize concentration Normalize concentration Figure 2.4: Steay state concentrations of the complex C an of A as functions of the concentration of B. As an exercise (Exercise 2.2), the reaer can verify that if B bins to A as a complex of n copies of B, that is, B + B + + B k 1 B n, B n + A a C, A +C = A tot, k2 then we have that the expressions of C an A change to C = A totb n /(K m K ) 1 + B n /(K m K ), A = A tot 1 + B n /(K m K ). In this case, we say that the bining of B to A is cooperative with cooperativity n. Figure 2.4 shows the above functions, which are often referre to as Hill functions an n is calle the Hill coefficient. Another type of cooperative bining is when a species R can bin A only after another species B has boun A. In this case, the reactions are given by B + A a C, R + C a C, A +C +C = A tot. Proceeing as above by writing the ODE moel an equating the time erivatives to zero to obtain the equilibrium, we obtain the equilibrium relations C = 1 B(A tot C C ), C = 1 K K K R(A tot C C ). By solving this system of two equations for the unknowns C an C, we obtain C = A tot (B/K )(R/K ) 1 + (B/K ) + (B/K )(R/K ), C = A tot (B/K ) 1 + (B/K ) + (B/K )(R/K ).

15 42 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES In the case in which B woul bin cooperatively with other copies of B with cooperativity n, the above expressions become C A tot (B n /K m K )(R/K = ) 1 + (B n /K m K )(R/K ) + (Bn /K m K ), A tot (B n /K m K ) C = 1 + (B n /K m K )(R/K ) + (Bn /K m K ). Competitive bining reaction. Finally, consier the case in which two species B a an B r both bin to A competitively, that is, they cannot be boun to A at the same time. Let C a be the complex forme between B a an A an let C r be the complex forme between B r an A. Then, we have the following reactions B a + A a C a, B r + A a C r, A +C a +C r = A tot, for which we can write the ifferential equation moel as C a = ab a (A tot C a C r ) C a, C r By setting the time erivatives to zero, we obtain so that = a B r (A tot C a C r ) C r. C a (ab a + ) = ab a (A tot C r ), C r (a B r + ) = a B r (A tot C a ), C r = B ( r(a tot C a ) B r + K, C a B a + K from which we finally etermine that C a = A tot (B a /K ) 1 + (B a /K ) + (B r /K ), C r = B ab r B r + K ) = B a ( K B r + K A tot (B r /K ) 1 + (B a /K ) + (B r /K ). ) A tot, In this erivation, we have assume that both B a an B r bin A as monomers. If they were bining as imers, the reaer shoul verify that they woul appear in the final expressions with a power of two (see Exercise 2.3). Note also that in this erivation we have assume that the bining is competitive, that is, B a an B r cannot simultaneously bin to A. If they can bin simultaneously to A, we have to inclue another complex comprising B a,b r an A. Denoting this new complex by C, we must a the two aitional reactions ā C a + B r C, C r + B a ā C,

16 2.1. MODELING CHEMICAL REACTIONS 43 an we shoul moify the conservation law for A to A tot = A + C a + C r + C. The reaer can verify that in this case a mixe term B r B a appears in the equilibrium expressions (see Exercise 2.4). Enzymatic reaction. A general enzymatic reaction can be written as E + S a C k E + P, in which E is an enzyme, S is the substrate to which the enzyme bins to form the complex C = ES, an P is the prouct resulting from the moification of the substrate S ue to the bining with the enzyme E. Here, a an are the association an issociation rate constants as before, an k is the catalytic rate constant. Enzymatic reactions are very common an inclue phosphorylation as we have seen in Example 2.1 an as we will see in more etail in the sequel. The corresponing ODE moel is given by S C = ae S + C, E P = ae S + C + kc, = ae S ( + k)c, = kc. The total enzyme concentration is usually constant an enote by E tot, so that E +C = E tot. Substituting E = E tot C in the above equations, we obtain S = a(e tot C) S + C, E = a(e tot C) S + C + kc, C = a(e tot C) S ( + k)c, P = kc. This system cannot be solve analytically, therefore, assumptions must be use in orer to reuce it to a simpler form. Michaelis an Menten assume that the conversion of E an S to C an vice versa is much faster than the ecomposition of C into E an P. Uner this assumption an letting the initial concentration S (0) be sufficiently large (see Example 3.12), C immeiately reaches its steay state value (while P is still changing). This approximation is calle the quasi-steay state approximation an the mathematical conitions on the parameters that justify it will be ealt with in Section 3.5. The steay state value of C is given by solving a(e tot C)S ( + k)c = 0 for C, which gives C = E tots, with K m = + k S + K m a, in which the constant K m is calle the Michaelis-Menten constant. Letting V max = ke tot, the resulting kinetics P = k E tots S = V max (2.9) S + K m S + K m

17 44 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Prouction rate, P/ First-orer Zero-orer Substrate concentration, S Prouct concentration, P K m = 1 K m = 0.5 K m = Time (s) (a) Prouction rate (b) Time plots Figure 2.5: Enzymatic reactions. (a) Transfer curve showing theprouctionrateforp as a function of substrate concentration for K m = 1. (b) Time plots of prouct P(t) for ifferent values of the K m.intheplotss tot = 1anV max = 1. are calle Michaelis-Menten kinetics. The constant V max is calle the maximal velocity (or maximal flux) of moification an it represents the maximal rate that can be obtaine when the enzyme is completely saturate by the substrate. The value of K m correspons to the value of S that leas to a half-maximal value of the prouction rate of P. When the enzyme complex can be neglecte with respect to the total substrate amount S tot, we have that S tot = S + P +C S + P, so that the above equation can be also rewritten as P = V max(s tot P). (S tot P) + K m When K m S tot an the substrate has not yet been all converte to prouct, that is, S K m, we have that the rate of prouct formation becomes approximately P/ V max, which is the maximal spee of reaction. Since this rate is constant an oes not epen on the reactant concentrations, it is usually referre to as zeroorer kinetics. In this case, the system is sai to operate in the zero-orer regime. If instea S K m, the rate of prouct formation becomes P/ V max /K m S, which is linear with the substrate concentration S. This prouction rate is referre to as first-orer kinetics an the system is sai to operate in the first-orer regime (see Figure 2.5). 2.2 Transcription an translation In this section we consier the processes of transcription an translation, using the moeling techniques escribe in the previous section to capture the funamental ynamic behavior. Moels of transcription an translation can be one at a variety of levels of etail an which moel to use epens on the questions that one

18 2.2. TRANSCRIPTION AND TRANSLATION 45 wants to consier. We present several levels of moeling here, starting with a fairly etaile set of reactions escribing transcription an translation an ening with highly simplifie orinary ifferential equation moels that can be use when we are only intereste in average prouction rate of mrna an proteins at relatively long time scales. The central ogma: Prouction of proteins The genetic material insie a cell, encoe in its DNA, governs the response of a cell to various conitions. DNA is organize into collections of genes, with each gene encoing a corresponing protein that performs a set of functions in the cell. The activation an repression of genes are etermine through a series of complex interactions that give rise to a remarkable set of circuits that perform the functions require for life, ranging from basic metabolism to locomotion to procreation. Genetic circuits that occur in nature are robust to external isturbances an can function in a variety of conitions. To unerstan how these processes occur (an some of the ynamics that govern their behavior), it will be useful to present a relatively etaile escription of the unerlying biochemistry involve in the prouction of proteins. DNA is a ouble strane molecule with the irection of each stran specifie by looking at the geometry of the sugars that make up its backbone. The complementary strans of DNA are compose of a sequence of nucleoties that consist of a sugar molecule (eoxyribose) boun to one of four bases: aenine (A), cytocine (C), guanine (G) an thymine (T). The coing region (by convention the top row of a DNA sequence when it is written in text form) is specifie from the 5 en of the DNA to the 3 en of the DNA. (The 5 an 3 refer to carbon locations on the eoxyribose backbone that are involve in linking together the nucleoties that make up DNA.) The DNA that encoes proteins consists of a promoter region, regulator regions (escribe in more etail below), a coing region an a termination region (see Figure 2.6). We informally refer to this entire sequence of DNA as a gene. Expression of a gene begins with the transcription of DNA into mrna by RNA polymerase, as illustrate in Figure 2.7. RNA polymerase enzymes are present in the nucleus (for eukaryotes) or cytoplasm (for prokaryotes) an must localize an bin to the promoter region of the DNA template. Once boun, the RNA polymerase opens the ouble strane DNA to expose the nucleoties that make up the sequence. This reaction, calle isomerization, is sai to transform the RNA polymerase an DNA from a close complex to an open complex. After the open complex is forme, RNA polymerase begins to travel own the DNA stran an constructs an mrna sequence that matches the 5 to 3 sequence of the DNA to which it is boun. By convention, we number the first base pair that is transcribe as +1 an the base pair prior to that (which is not transcribe) is labele as -1. The promoter region is often shown with the -10 an -35 regions inicate, since these

19 46 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES RNA polymerase RNA RBS AUG UAA 5' 3' Start coon Stop coon Transcription 5' 3' TTACA TATGTT AGGAGGT AATGT ATACAA TCCTCCA Promoter ATG TAC DNA TAA ATT Terminator 3' 5' Figure 2.6: Geometric structure of DNA. The layout of the DNA is shown at the top. RNA polymerase bins to the promoter region of the DNA an transcribes the DNA starting at the +1 siteancontinuingtotheterminationsite.thetranscribe mrna stran has the ribosome bining site (RBS) where the ribosomes bin, the start coon where translation starts an the stop coon where translation ens. regions contain the nucleotie sequences to which the RNA polymerase enzyme bins (the locations vary in ifferent cell types, but these two numbers are typically use). The RNA stran that is prouce by RNA polymerase is also a sequence of nucleoties with a sugar backbone. The sugar for RNA is ribose instea of eoxyribose an mrna typically exists as a single strane molecule. Another ifference is that the base thymine (T) is replace by uracil (U) in RNA sequences. RNA polymerase prouces RNA one base pair at a time, as it moves from in the 5 to 3 irection along the DNA coing region. RNA polymerase stops transcribing DNA when it reaches a termination region (orterminator) on the DNA. This termination region consists of a sequence that causes the RNA polymerase to unbin from the DNA. The sequence is not conserve across species an in many cells the termination sequence is sometimes leaky, so that transcription will occasionally occur across the terminator. Once the mrna is prouce, it must be translate into a protein. This process is slightly ifferent in prokaryotes an eukaryotes. In prokaryotes, there is a region of the mrna in which the ribosome (a molecular complex consisting of both proteins an RNA) bins. This region, calle the ribosome bining site (RBS), has some variability between ifferent cell species an between ifferent genes in a given cell. The Shine-Dalgarno sequence, AGGAGG, is the consensus sequence for the RBS. (A consensus sequence is a pattern of nucleoties that implements a given function across multiple organisms; it is not exactly conserve, so some variations in the sequence will be present from one organism to another.) In eukaryotes, the RNA must unergo several aitional steps before it is trans-

20 2.2. TRANSCRIPTION AND TRANSLATION 47 Promoter Terminator 5' 3' 3' 5' Accessory factors Promoter complex Initiation RNA polymerase Nascent transcript 5' Elongation 3' Elongation Termination Full-length transcript Figure 2.7: Prouction of messenger RNA from DNA. RNA polymerase, along with other accessory factors, bins to the promoter region of the DNA an then opens the DNA to begin transcription (initiation). As RNA polymerase moves own the DNA in the transcription elongation complex (TEC), it prouces an RNA transcript (elongation), which is later translate into a protein. The process ens when the RNA polymerase reaches the terminator (termination). Figure aapte from Courey [20]. late. The RNA sequence that has been create by RNA polymerase consists of introns that must be splice out of the RNA (by a molecular complex calle the spliceosome), leaving only the exons, which contain the coing region for the pro-

21 48 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES tein. The term pre-mrna is often use to istinguish between the raw transcript an the splice mrna sequence, which is calle mature mrna. In aition to splicing, the mrna is also moifie to contain a poly(a) (polyaenine) tail, consisting of a long sequence of aenine (A) nucleoties on the 3 en of the mrna. This processe sequence is then transporte out of the nucleus into the cytoplasm, where the ribosomes can bin to it. Unlike prokaryotes, eukaryotes o not have a well-efine ribosome bining sequence an hence the process of the bining of the ribosome to the mrna is more complicate. The Kozak sequence, A/GCCACCAUGG, is the rough equivalent of the ribosome bining site, where the unerline AUG is the start coon (escribe below). However, mrna lacking the Kozak sequence can also be translate. Once the ribosome is boun to the mrna, it begins the process of translation. Proteins consist of a sequence of amino acis, with each amino aci specifie by a coon that is use by the ribosome in the process of translation. Each coon consists of three base-pairs an correspons to one of the twenty amino acis or a stop coon. The ribosome translates each coon into the corresponing amino aci using transfer RNA (trna) to integrate the appropriate amino aci (which bins to the trna) into the polypeptie chain, as shown in Figure 2.8. The start coon (AUG) specifies the location at which translation begins, as well as coing for the amino aci methionine (a moifie form is use in prokaryotes). All subsequent coons are translate by the ribosome into the corresponing amino aci until it reaches one of the stop coons (typically UAA, UAG an UGA). The sequence of amino acis prouce by the ribosome is a polypeptie chain that fols on itself to form a protein. The process of foling is complicate an involves a variety of chemical interactions that are not completely unerstoo. Aitional post-translational processing of the protein can also occur at this stage, until a fole an functional protein is prouce. It is this molecule that is able to bin to other species in the cell an perform the chemical reactions that unerlie the behavior of the organism. The maturation time of a protein is the time require for the polypeptie chain to fol into a functional protein. Each of the processes involve in transcription, translation an foling of the protein takes time an affects the ynamics of the cell. Table 2.1 shows representative rates of some of the key processes involve in the prouction of proteins. In particular, the issociation constant of RNA polymerase from the DNA promoter has a wie range of values epening on whether the bining is enhance by activators (as we will see in the sequel), in which case it can take very low values. Similarly, the issociation constant of transcription factors with DNA can be very low in the case of specific bining an substantially larger for non-specific bining. It is important to note that each of these steps is highly stochastic, with molecules bining together base on some propensity that epens on the bining energy but also the other molecules present in the cell. In aition, although we have escribe everything as a sequential process, each of the steps of tran-

22 TRANSCRIPTION AND TRANSLATION 5' DNA Translation Amino aci Growing amino aci chain Transcription trna Transport to cytoplasm RNA 3' trna ocking Nucleus trna leaving Coon Ribosome mrna Cytoplasm Figure 2.8: Translation is the process of translating the sequence of a messenger RNA (mrna) molecule to a sequence of amino acis uring protein synthesis. The genetic coe escribes the relationship between the sequence of base pairs in a gene an the corresponing amino aci sequence that it encoes. In the cell cytoplasm, the ribosome reas the sequence of the mrna in groups of three bases to assemble the protein. Figure an caption courtesy the National Human Genome Research Institute. Table 2.1: Rates of core processes involve in the creation of proteins from DNA in E. coli. Process mrna transcription rate Protein translation rate Maturation time (fluorescent proteins) mrna half-life E. coli cell ivision time Yeast cell ivision time Protein half-life Protein iffusion along DNA RNA polymerase issociation constant Open complex formation kinetic rate Transcription factor issociation constant Characteristic rate bp/s aa/s 6 60 min 100 s min min s up to 104 bp/s ,000 nm 0.02 s ,000 nm Source [13] [13] [13] [103] [13] [13] [103] [78] [13] [13] [13]

23 50 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES scription, translation an foling are happening simultaneously. In fact, there can be multiple RNA polymerases that are boun to the DNA, each proucing a transcript. In prokaryotes, as soon as the ribosome bining site has been transcribe, the ribosome can bin an begin translation. It is also possible to have multiple ribosomes boun to a single piece of mrna. Hence the overall process can be extremely stochastic an asynchronous. Reaction moels The basic reactions that unerlie transcription inclue the iffusion of RNA polymerase from one part of the cell to the promoter region, bining of an RNA polymerase to the promoter, isomerization from the close complex to the open complex, an finally the prouction of mrna, one base-pair at a time. To capture this set of reactions, we keep track of the various forms of RNA polymerase accoring to its location an state: RNAP c represents RNA polymerase in the cytoplasm, RNAP p represents RNA polymerase in the promoter region, an RNAP is RNA polymerase non-specifically boun to DNA. We must similarly keep track of the state of the DNA, to ensure that multiple RNA polymerases o not bin to the same section of DNA. Thus we can write DNA p for the promoter region, DNA i for the ith section of the gene of interest an DNA t for the termination sequence. We write RNAP : DNA to represent RNA polymerase boun to DNA (assume close) an RNAP : DNA o to inicate the open complex. Finally, we must keep track of the mrna that is prouce by transcription: we write mrna i to represent an mrna stran of length i an assume that the length of the gene of interest is N. Using these various states of the RNA polymerase an locations on the DNA, we can write a set of reactions moeling the basic elements of transcription as Bining to DNA: Diffusion along DNA: RNAP c RNAP, RNAP RNAP p, Bining to promoter: RNAP p + DNA p RNAP : DNA p, Isomerization: RNAP : DNA p RNAP : DNA o, Start of transcription: RNAP : DNA o RNAP : DNA 1 + DNA p, mrna creation: RNAP : DNA 1 RNAP : DNA 2 : mrna 1, Elongation: RNAP : DNA i+1 : mrna i RNAP : DNA i+2 : mrna i+1, Bining to terminator: RNAP:DNA N : mrna N 1 RNAP : DNA t + mrna N, Termination: RNAP : DNA t RNAP c, Degraation: mrna N. (2.10)

24 2.2. TRANSCRIPTION AND TRANSLATION 51 Note that at the start of transcription we release the promoter region of thedna, thus allowing a secon RNA polymerase to bin to the promoter while the first RNA polymerase is still transcribing the gene. This allows the same DNA stran to be transcribe by multiple RNA polymerase at the same time. The species RNAP : DNA i+1 : mrna i represents RNA polymerases boun at the (i + 1)th section of DNA with an elongating mrna stran of length i attache to it. Upon bining to the terminator region, the RNA polymerase releases the full mrna stran mrna N. This mrna has the ribosome bining site at which ribosomes can bin to start translation. The main ifference between prokaryotes an eukaryotes is that in eukaryotes the RNA polymerase remains in the nucleus an the mrna N must be splice an transporte to the cytoplasm before ribosomes can start translation. As a consequence, the start of translation can occur only after mrna N has been prouce. For simplicity of notation, we assume here that the entire mrna stran shoul be prouce before ribosomes can start translation. In the procaryotic case, instea, translation can start even for an mrna stran that is still elongating (see Exercise 2.6). A similar set of reactions can be written to moel the process of translation. Here we must keep track of the bining of the ribosome to the ribosome bining site (RBS) of mrna N, translation of the mrna sequence into a polypeptie chain, an foling of the polypeptie chain into a functional protein. Specifically, we must keep track of the various states of the ribosome boun to ifferent coons on the mrna stran. We thus let Ribo : mrna RBS enote the ribosome boun to the ribosome bining site of mrna N, Ribo : mrna AAi the ribosome boun to the ith coon (corresponing to an amino aci, inicate by the superscript AA), Ribo : mrna start an Ribo : mrna stop the ribosome boun to the start an stop coon, respectively. We also let PPC i enote the polypeptie chain consisting of i amino acis. Here, we assume that the protein of interest has M amino acis. The reactions escribing translation can then be written as Bining to RBS: Ribo + mrna N Ribo : mrna RBS, Start of translation: Ribo : mrna RBS Ribo : mrna start + mrna N, Polypeptie chain creation: Ribo : mrna start Ribo : mrna AA2 : PPC 1, Elongation, i = 1,...,M: Stop coon: Release of mrna: Foling: Ribo : mrna AA(i+1) : PPC i Ribo : mrna AAM : PPC M 1 Ribo : mrna stop Ribo, PPC M protein, Degraation: protein. Ribo : mrna AA(i+2) : PPC i+1, Ribo : mrna stop + PPC M, (2.11)

25 52 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES As in the case of transcription, we see that these reactions allow multiple ribosomes to translate the same piece of mrna by freeing up mrna N. After M amino acis have been chaine together, the M-long polypeptie chain PPC M is release, which then fols into a protein. As complex as these reactions are, they o not irectly capture a number of physical phenomena such as ribosome queuing, wherein ribosomes cannot pass other ribosomes that are ahea of them on the mrna chain. Aitionally, we have not accounte for the existence an effects of the 5 an 3 untranslate regions (UTRs) of a gene an we have also left out various error correction mechanisms in which ribosomes can step back an release an incorrect amino aci that has been incorporate into the polypeptie chain. We have also left out the many chemical species that must be present in orer for a variety of the reactions to happen (NTPs for mrna prouction, amino acis for protein prouction, etc.). Incorporation of these effects requires aitional reactions that track the many possible states of the molecular machinery that unerlies transcription an translation. For more etaile moels of translation, the reaer is referre to [3]. When the etails of the isomerization, start of transcription (translation), elongation, an termination are not relevant for the phenomenon to be stuie, the transcription an translation reactions are lumpe into much simpler reuce reactions. For transcription, these reuce reactions take the form: RNAP + DNA p RNAP:DNA p, RNAP:DNA p mrna + RNAP + DNA p, mrna, (2.12) in which the secon reaction lumps together isomerization, start of transcription, elongation, mrna creation, an termination. Similarly, for the translation process, the reuce reactions take the form: Ribo + mrna Ribo:mRNA, Ribo:mRNA protein + mrna + Ribo, Ribo:mRNA Ribo, protein, (2.13) in which the secon reaction lumps the start of translation, elongation, foling, an termination. The thir reaction moels the fact that mrna can also be egrae when boun to ribosomes when the ribosome bining site is left free. The process of mrna egraation occurs through RNase enzymes bining to the ribosome bining site an cleaving the mrna stran. It is known that the ribosome bining site cannot be both boun to the ribosome an to the RNase [68]. However, the species Ribo : mrna is a lumpe species encompassing configurations in which ribosomes are boun on the mrna stran but not on the ribosome bining site. Hence, we also allow this species to be egrae by RNase.

26 2.2. TRANSCRIPTION AND TRANSLATION 53 Reaction rate equations Given a set of reactions, the various stochastic processes that unerlie etaile moels of transcription an translation can be specifie using the stochastic moeling framework escribe briefly in the previous section. In particular, using either moels of bining energy or measure rates, we can construct propensity functions for each of the many reactions that lea to prouction of proteins, incluing the motion of RNA polymerase an the ribosome along DNA an RNA. For many problems in which the etaile stochastic nature of the molecular ynamics of the cell are important, these moels are the most relevant an they are covere in some etail in Chapter 4. Alternatively, we can move to the reaction rate formalism an moel the reactions using ifferential equations. To o so, we must compute the various reaction rates, which can be obtaine from the propensity functions or measure experimentally. In moving to this formalism, we approximate the concentrations of various species as real numbers (though this may not be accurate for some species that exist at low molecular counts in the cell). Despite these approximations, in many situations the reaction rate equations are sufficient, particularly if we are intereste in the average behavior of a large number of cells. In some situations, an even simpler moel of the transcription, translation an foling processes can be utilize. Let the active mrna be the mrna that is available for translation by the ribosome. We moel its concentration through a simple time elay of length τ m that accounts for the transcription of the ribosome bining site in prokaryotes or splicing an transport from the nucleus in eukaryotes. If we assume that RNA polymerase bins to DNA at some average rate (which inclues both the bining an isomerization reactions) an that transcription takes some fixe time (epening on the length of the gene), then the process of transcription can be escribe using the elay ifferential equation m P = α µm P δm P, m P (t) = e µτm m P (t τ m ), (2.14) where m P is the concentration of mrna for protein P, m P is the concentration of active mrna, α is the rate of prouction of the mrna for protein P, µ is the growth rate of the cell (which results in ilution of the concentration) an δ is the rate of egraation of the mrna. Since the ilution an egraation terms are of the same form, we will often combine these terms in the mrna ynamics an use a single coefficient δ = µ + δ. The exponential factor in the secon expression in equation (2.14) accounts for ilution ue to the change in volume of the cell, where µ is the cell growth rate. The constants α an δ capture the average rates of prouction an ecay, which in turn epen on the more etaile biochemical reactions that unerlie transcription. Once the active mrna is prouce, the process of translation can be escribe via a similar orinary ifferential equation that escribes the prouction of a func-

27 54 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES tional protein: P = κm P γp, P f (t) = e µτ f P(t τ f ). (2.15) Here P represents the concentration of the polypeptie chain for the protein, an P f represents the concentration of functional protein (after foling). The parameters that govern the ynamics are κ, the rate of translation of mrna; γ, the rate of egraation an ilution of P; an τ f, the time elay associate with foling an other processes require to make the protein functional. The exponential term again accounts for ilution ue to cell growth. The egraation an ilution term, parameterize by γ, captures both the rate at which the polypeptie chain is egrae an the rate at which the concentration is ilute ue to cell growth. It will often be convenient to write the ynamics for transcription an translation in terms of the functional mrna an functional protein. Differentiating the expression for m P, we see that m P (t) = e m µτm P (t τ m ) = e µτm( α δm P (t τ m ) ) (2.16) = ᾱ δm P (t), where ᾱ = e µτm α. A similar expansion for the active protein ynamics yiels P f (t) = κm P (t τ f ) γp f (t), (2.17) where κ = e µτ f κ. We shall typically use equations (2.16) an (2.17) as our (reuce) escription of protein foling, ropping the superscript f an overbars when there is no risk of confusion. Also, in the presence of ifferent proteins, we will attach subscripts to the parameters to enote the protein to which they refer. In many situations the time elays escribe in the ynamics of protein prouction are small compare with the time scales at which the protein concentration changes (epening on the values of the other parameters in the system). In such cases, we can simplify our moel of the ynamics of protein prouction even further an write m P P = α δm P, = κm P γp. (2.18) Note that we here have roppe the superscripts an f since we are assuming that all mrna is active an proteins are functional an roppe the overbar on α an κ since we are assuming the time elays are negligible. The value of α increases with the strength of the promoter while the value of κ increases with the strength of the ribosome bining site. These strengths, in turn, can be affecte by changing the specific base-pair sequences that constitute the promoter RNA polymerase bining region an the ribosome bining site. Finally, the simplest moel for protein prouction is one in which we only keep track of the basal rate of prouction of the protein, without incluing the mrna

28 2.3. TRANSCRIPTIONAL REGULATION 55 ynamics. This essentially amounts to assuming the mrna ynamics reach steay state quickly an replacing the first ifferential equation in (2.18) with its equilibrium value. This is often a goo assumption as mrna egration is usually about 100 times faster than protein egraation (see Table 2.1). Thus we obtain P = β γp, β := κα δ. This moel represents a simple first-orer, linear ifferential equation for the rate of prouction of a protein. In many cases this will be a sufficiently goo approximate moel, although we will see that in some cases it is too simple to capture the observe behavior of a biological circuit. 2.3 Transcriptional regulation The operation of a cell is governe in part by the selective expression of genes in the DNA of the organism, which control the various functions the cell is able to perform at any given time. Regulation of protein activity is a major component of the molecular activities in a cell. By turning genes on an off, an moulating their activity in more fine-graine ways, the cell controls its many metabolic pathways, respons to external stimuli, ifferentiates into ifferent cell types as it ivies, an maintains the internal state of the cell require to sustain life. The regulation of gene expression an protein activity is accomplishe through a variety of molecular mechanisms, as iscusse in Section 1.2 an illustrate in Figure 2.9. At each stage of the processing from a gene to a protein, there are potential mechanisms for regulating the prouction processes. The remainer of this section will focus on transcriptional control an the next section on selecte mechanisms for controlling protein activity. We will focus on prokaryotic mechanisms. Transcriptional regulation of protein prouction The simplest forms of transcriptional regulation are repression an activation, both controlle through proteins calle transcription factors. In the case of repression, the presence of a transcription factor (often a protein that bins near the promoter) turns off the transcription of the gene an this type of regulation is often calle negative regulation or own regulation. In the case of activation (or positive regulation), transcription is enhance when an activator protein bins to the promoter site (facilitating bining of the RNA polymerase). Repression. A common mechanism for repression is that a protein bins to a region of DNA near the promoter an blocks RNA polymerase from bining. The region of DNA to which the repressor protein bins is calle an operator region (see Figure 2.10a). If the operator region overlaps the promoter, then the presence of a protein at the promoter can block the DNA at that location an transcription

29 56 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES DNA Growing polypeptie chain Ribosome mrna Transcriptional control RNA polymerase Translation control Protein mrna processing control Protein activity control mrna Nucleus mrna Cytosol Active protein Growing polypeptie chain mrna transport control Ribosome Protein egraation mrna Figure 2.9: Regulation of proteins. Transcriptional control inclues mechanisms to tune the rate at which mrna is prouce from DNA, while translation controlincluesmechanisms to tune the rate at which the protein polypeptie chain is prouce from mrna. Protein activity control encompasses many processes, such as phosphorylation, methylation, an allosteric moification. Figure aapte from Phillips, Konev an Theriot [78]. cannot initiate. Repressor proteins often bin to DNA as imers or pairs of imers (effectively tetramers). Figure 2.10b shows some examples of repressors boun to DNA. A relate mechanism for repression is DNA looping. In this setting, two repressor complexes (often imers) bin in ifferent locations on the DNA an then bin to each other. This can create a loop in the DNA an block the ability of RNA polymerase to bin to the promoter, thus inhibiting transcription. Figure 2.11 shows an example of this type of repression, in the lac operon. (An operon is a set of genes

30 2.3. TRANSCRIPTIONAL REGULATION 57 Promoter Genes Operator RNA polymerase Repressor Genes on Low repressor concentration Genes off High repressor concentration (a) Repression of gene expression (b) Examples of repressors Figure 2.10: Repression of gene expression. A repressor protein bins to operator sites on the gene promoter an blocks the bining of RNA polymerase to the promoter, so that the gene is off. FiguresaaptefromPhillips,KonevanTheriot[78]. Copyright 2009 from Physical Biology of the Cell by Phillips et al. Reprouce by permission of Garlan Science/Taylor & Francis LLC. that is uner control of a single promoter.) Activation. The process of activation of a gene requires that an activator protein be present in orer for transcription to occur. In this case, the protein must work to either recruit or enable RNA polymerase to begin transcription. The simplest form of activation involves a protein bining to the DNA near the promoter in such a way that the combination of the activator an the promoter sequence bin RNA polymerase. Figure 2.12 illustrates the basic concept. Another mechanism for activation of transcription, specific to prokaryotes, is the use of sigma factors. Sigma factors are part of a moular set of proteins that bin to RNA polymerase an form the molecular complex that performs transcription. Different sigma factors enable RNA polymerase to bin to ifferent promoters, so the sigma factor acts as a type of activating signal for transcription. Table 2.2 lists some of the common sigma factors in bacteria. One of the uses of sigma factors is to prouce certain proteins only uner special conitions, such as when the cell unergoes heat shock. Another use is to control the timing of the expression of certain genes, as illustrate in Figure 2.13.

31 58 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES 5 nm (a) DNA looping (b) lac repressor Figure 2.11: Repression via DNA looping. A repressor protein canbinsimultaneouslyto two DNA sites ownstream of the start of transcription, thus creating a loop that prevents RNA polymerase from transcribing the gene. Figures aapte from Phillips, Konev an Theriot [78]. Copyright 2009 from Physical Biology of the Cell by Phillips et al. Reprouce by permission of Garlan Science/Taylor & Francis LLC. Inucers. A feature that is present in some types of transcription factors is the existence of an inucer molecule that combines with the protein to either activate or inactivate its function. A positive inucer is a molecule that must be present in orer for repression or activation to occur. A negative inucer is one in which the presence of the inucer molecule blocks repression or activation, either by changing the shape of the transcription factor protein or by blocking active sites on the protein that woul normally bin to the DNA. Figure 2.14 summarizes the various possibilities. Common examples of repressor-inucer pairs inclue laci an lactose (or IPTG), an tetr an atc. Lactose/IPTG an atc are both negative inucers, so their presence causes the otherwise represse gene to be expresse. An example of a positive inucer is cyclic AMP (camp), which acts as a positive inucer for the CAP activator. Combinatorial promoters. In aition to promoters that can take either a repressor or an activator as the sole input transcription factor, there are combinatorial promoters that can take both repressors an activators as input transcription factors. This allows genes to be switche on an off base on more complex conitions, represente by the concentrations of two or more activators or repressors. Table 2.2: Sigma factors in E. coli [2]. Sigma factor σ 70 σ 32 σ 38 σ 28 σ 24 Promoters recognize most genes genes associate with heat shock genes involve in stationary phase an stress response genes involve in motility an chemotaxis genes ealing with misfole proteins in the periplasm

32 2.3. TRANSCRIPTIONAL REGULATION 59 Activator Ahesive interaction RNA polymerase (a) Activation mechanism (b) Examples of activators Figure 2.12: Activation of gene expression. (a) Conceptual operation of an activator. The activator bins to DNA upstream of the gene an attracts RNA polymerase to the DNA stran. (b) Examples of activators: catabolite activator protein (CAP), p53 tumor suppressor, zinc finger DNA bining omain an leucine zipper DAN bining omain. Figures aapte from Phillips, Konev an Theriot [78]. Copyright 2009 from Physical Biology of the Cell by Phillips et al. Reprouce by permission of Garlan Science/Taylor & Francis LLC. RNA polymerase with bacterial sigma factor RNA polymerase with viral sigma factor Viral DNA Early genes 28 Mile genes 34 Late genes Figure 2.13: Use of sigma factors to control the timing of gene expressioninabacterial virus. Early genes are transcribe by RNA polymerase boun to bacterial sigma factors. One of the early genes, calle 28, encoes a sigma-like factor that bins to RNA polymerase an allow it to transcribe mile genes, which in turn prouce another sigma-like factor that allows RNA polymerase to transcribe late genes. These late genes prouce proteins that form a coat for the viral DNA an lyse the cell. Figure aapte from Alberts et al. [2].

33 60 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Ligan bins to remove regulatory protein from DNA Boun repressor protein prevents transcription Boun repressor protein Aition of ligan switches gene on by removing repressor protein Gene off Ligan bins to remove regulatory protein from DNA Boun activator protein Boun activator protein promotes transcription Aition of ligan switches gene off by removing activator protein RNA polymerase Protein Gene on mrna 5' 3' Ligan bins to allow regulatory protein to bin to DNA Removal of ligan switches gene on by removing repressor protein (a) Repressor Inactive repressor Gene off Ligan bins to allow regulatory protein to bin to DNA Removal of ligan switches gene off by removing activator protein (b) Activator Protein Gene on mrna 5' 3' Figure 2.14: Effects of inucers. (a) In the case of repressors, a negative inucer bins to the repressor making it unbin DNA, thus enabling transcription. A positive inucer, by contrast, activates the repressor allowing it to bin DNA. (b) In the case of activators, a negative inucer bins to the activator making it unbin DNA, thus preventing transcription. A positive inucer instea enables the activator to bin DNA, allowing transcription. Figures aapte from Alberts et al. [2]. Figure 2.15 shows one of the classic examples, a promoter for the lac system. In the lac system, the expression of genes for metabolizing lactose are uner the control of a single (combinatorial) promoter. CAP, which is positively inuce by camp, acts as an activator an LacI (also calle Lac repressor ), which is negatively inuce by lactose, acts as a repressor. In aition, the inucer camp is expresse only when glucose levels are low. The resulting behavior is that the proteins for metabolizing lactose are expresse only in conitions where there is no glucose (so CAP is active) an lactose is present. More complicate combinatorial promoters can also be use to control transcription in two ifferent irections, an example that is foun in some viruses. Antitermination. A final metho of activation in prokaryotes is the use of antitermination. The basic mechanism involves a protein that bins to DNA an eactivates a site that woul normally serve as a termination site for RNA polymerase. Aitional genes are locate ownstream from the termination site, but without a promoter region. Thus, in the absence of the antiterminator protein, these genes are not expresse (or expresse with low probability). However, when the antitermination protein is present, the RNA polymerase maintains (or regains) its contact with

34 2.3. TRANSCRIPTIONAL REGULATION 61 CAPbining site RNA polymerasebining site (promoter) Start site for RNA synthesis Operator lacz gene + Glucose + Lactose Nucleotie pairs Operon off because CAP not boun + Glucose Lactose CAP Repressor Operon off both because Lac repressor boun an because CAP not boun Glucose Lactose RNA polymerase Operon off because Lac repressor boun Glucose + Lactose mrna 5' 3' Operon on Figure 2.15: Combinatorial logic for the lac operator. The CAP-bining site an the operator in the promoter can be both boun by CAP (activator) an by LacI(Lacrepressor), respectively. The only configuration in which RNA polymerasecanbinhepromoteran start transcription is where CAP is boun but LacI is not boun. Figure aapte from Phillips, Konev an Theriot [78]. the DNA an expression of the ownstream genes is enhance. In this way, antitermination allows ownstream genes to be regulate by repressing premature termination. An example of an antitermination protein is the protein N in phage λ, which bins to a region of DNA labele nut (for N utilization), as shown in Figure 2.16 [37]. Reaction moels We can capture the molecular interactions responsible for transcriptional regulation by moifying the RNA polymerase bining reactions in equation (2.10) to inclue the bining of the repressor or activator to the promoter. For a repressor (Rep), we have to a a reaction that represents the repressor boun to the promoter DNA p : Repressor bining: DNA p + Rep DNA:Rep.

35 62 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES nutl nutr λ pl N + N pr Figure 2.16: Antitermination. Protein N bins to DNA regions labele nut, enabling transcription of longer DNA sequences. Figure aapte from [37]. This reaction acts to sequester the DNA promoter site so that it is no longer available for bining by RNA polymerase. The strength of the repressor is reflecte in the reaction rate constants for the repressor bining reaction. Sometimes, the RNA polymerase can bin to the promoter even when the repressor is boun, usually with lower association rate constant. In this case, the repressor still allows some transcription even when boun to the promoter an the repressor is sai to be leaky. The moifications for an activator (Act) are a bit more complicate, since we have to moify the reactions to require the presence of the activator before RNA polymerase can bin the promoter. One possible mechanism, known as the recruitment moel, is given by Activator bining: RNAP bining w/ activator: Isomerization: DNA p + Act DNA p :Act, RNAP p + DNA p :Act RNAP:DNA p :Act, RNAP:DNA p :Act RNAP:DNA o :Act, Start of transcription: RNAP:DNA o :Act RNAP:DNA 1 + DNA p :Act. (2.19) In this moel, RNA polymerase cannot bin to the promoter unless the activator is alreay boun to it. More generally, one can moel both the enhance bining of the RNA polymerase to the promoter in the presence of the activator, as well as the possibility of bining without an activator. This translates into the aitional reaction RNAP p + DNA p RNAP:DNA p. The relative reaction rates etermine how strong the activator is an the leakiness of transcription in the absence of the activator. A ifferent moel of activation, calle allosteric activation, is one in which the RNA polymerase bining rate to DNA is not enhance by the presence of the activator boun to the promoter, but the open complex (an hence start of transcription) formation can occur only (is enhance) in the presence of the activator. A simplifie orinary ifferential equation moel of transcription in the presence of activators or repressors can be obtaine by accounting for the fact that transcription factors an RNAP bin to the DNA rapily when compare to other reactions, such as isomerization an elongation. As a consequence, we can make

36 2.3. TRANSCRIPTIONAL REGULATION 63 use of the reuce-orer moels that escribe the quasi-steay state concentrations of proteins boun to DNA as escribe in Section 2.1. We can consier the competitive bining case to moel a strong repressor that prevents RNA polymerase from bining to the DNA. In the sequel, we remove the superscripts p an from RNA polymerase to simplify notation. The quasi-steay state concentration of the complex of DNA promoter boun to the repressor will have the expression [DNA p [DNA] tot ([Rep]/K ) :Rep] = 1 + [Rep]/K + [RNAP]/K an the steay state amount of DNA promoter boun to the RNA polymerase will be given by [RNAP:DNA p ] = [DNA] tot([rnap]/k ) 1 + [RNAP]/K + [Rep]/K, in which K is the issociation constant of RNA polymerase from the promoter, while K is the issociation constant of Rep from the promoter, an [DNA] tot represents the total concentration of DNA. The free DNA promoter with RNA polymerase boun will allow transcription, while the complex DNA p : Rep will not allow transcription as it is not boun to RNA polymerase. Using the lumpe reactions (2.12) with reaction rate constant k f, this can be moele as [mrna] in which the prouction rate is given by = F([Rep]) δ[mrna], [DNA] tot ([RNAP]/K F([Rep]) = k ) f 1 + [RNAP]/K + [Rep]/K. If the repressor bins to the promoter with cooperativity n, the above expression becomes (see Section 2.1) F([Rep]) = k f [DNA] tot ([RNAP]/K ) 1 + [RNAP]/K + [Rep]n /(K m K ), in which K m is the issociation constant of the reaction of n molecules of Rep bining together. The function F is usually represente in the stanar Hill function form α F([Rep]) = 1 + ([Rep]/K) n, in which α an K are given by α = k f[dna] tot ([RNAP]/K ) 1 + ([RNAP]/K ), K = ( K m K (1 + ([RNAP]/K )) 1/n.

37 64 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Finally, if the repressor allows RNA polymerase to still bin to the promoter at a small rate (leaky repressor), the above expression can be moifie to take the form F([Rep]) = α 1 + ([Rep]/K) n + α 0, (2.20) in which α 0 is the basal expression rate when the promoter is fully represse, usually referre to as leakiness (see Exercise 2.8). To moel the prouction rate of mrna in the case in which an activator Act is require for transcription, we can consier the case in which RNA polymerase bins only when the activator is alreay boun to the promoter (recruitment moel). To simplify the mathematical erivation, we rewrite the reactions (2.19) involving the activator with the lumpe transcription reaction (2.12) into the following: DNA p + Act DNA p :Act, RNAP + DNA p :Act RNAP:DNA p :Act, RNAP:DNA p :Act k f mrna + RNAP + DNA p :Act, (2.21) in which the thir reaction lumps together isomerization, start of transcription, elongation an termination. The first an secon reactions fit the structure of the cooperative bining moel illustrate in Section 2.1. Also, since the thir reaction is much slower than the first two, the complex RNAP : DNA p : Act concentration can be well approximate at its quasi-steay state value. The expression of the quasi-steay state concentration was given in Section 2.1 in corresponence to the cooperative bining moel an takes the form [RNAP:DNA p :Act] = [DNA] tot([rnap]/k )([Act])/K ) 1 + ([Act]/K )(1 + [RNAP]/K ), in which K is the issociation constant of RNA polymerase with the complex of DNA boun to Act an K is the issociation constant of Act with DNA. When the activator Act bins to the promoter with cooperativity n, the above expression becomes [RNAP:DNA p :Act] = [DNA] tot([rnap][act] n )/(K K K m) 1 + ([Act] n /K K m )(1 + [RNAP]/K ), in which K m is the issociation constant of the reaction of n molecules of Act bining together. In orer to write the ifferential equation for the mrna concentration, we consier the thir reaction in (2.21) along with the above quasi-steay state expressions of [RNAP : DNA p : Act] to obtain [mrna] = F([Act]) δ[mrna],

38 2.3. TRANSCRIPTIONAL REGULATION 65 in which F([Act]) = k f [DNA] tot ([RNAP][Act] n )/(K K K m) 1 + ([Act] n /K K m )(1 + [RNAP]/K ) =: α([act]/k) n 1 + ([Act]/K) n, where α an K are implicitly efine. The right-han sie of this expression is in stanar Hill function form. If we assume that RNA polymerase can still bin to DNA even when the activator is not boun, we have an aitional basal expression rate α 0 so that the new form of the prouction rate is given by (see Exercise 2.9) F([Act]) = α([act]/k)n 1 + ([Act]/K) n + α 0. (2.22) As inicate earlier, many activators an repressors operate in the presence of inucers. To incorporate these ynamics in our escription, we simply have to a the reactions that correspon to the interaction of the inucer with the relevant protein. For a negative inucer, we can a a reaction in which the inucer bins the regulator protein an effectively sequesters it so that it cannot interact with the DNA. For example, a negative inucer operating on a repressor coul be moele by aing the reaction Rep + In Rep:In. Since the above reactions are very fast compare to transcription, they can be assume at the quasi-steay state. Hence, the free amount of repressor that can still bin to the promoter can be calculate by writing the ODE moel corresponing to the above reactions an by setting the time erivatives to zero. This yiels [Rep] = [Rep] tot, 1 + [In]/ K in which [Rep] tot = [Rep] + [Rep:In] is the total amount of repressor (boun an unboun to the inucer) an K is the issociation constant of In bining to Rep. This expression of the repressor concentration nees to be substitute in the expression of the prouction rate F([Rep]). Positive inucers can be hanle similarly, except now we have to moify the bining reactions to only work in the presence of a regulatory protein boun to an inucer. For example, a positive inucer on an activator woul have the moifie reactions Inucer bining: Activator bining: RNAP bining w/ activator: Isomerization: Act + In Act:In, DNA p + Act:In DNA p :Act:In, RNAP + DNA p :Act:In RNAP:DNA p :Act:In, RNAP:DNA p :Act:In RNAP:DNA o :Act:In, Start of transcription: RNAP:DNA o :Act:In RNAP:DNA 1 +DNA p :Act:In.

39 66 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES A A A m A m A m A (a) Unregulate (b) Negative autoregulation (c) Positive autoregulation Figure 2.17: Autoregulation of gene expression. In (a) the circuit is unregulate, while (b) shows negative autoregulation an (c) shows positive autoregulation. Hence, in the expression of the prouction rate F([Act]), we shoul substitute the concentration [Act:In] in place of [Act]. This concentration, in turn, is well approximate by its quasi-steay state value since bining reactions are much faster than isomerization an transcription, an can be obtaine as in the negative inucer case. Example 2.2 (Autoregulation of gene expression). Consier the circuits shown in Figure 2.17, representing an unregulate gene, a negatively autoregulate gene an a positively autoregulate gene. We want to moel the ynamics of the protein A starting from zero initial conitions for the three ifferent cases to unerstan how the three ifferent circuit topologies affect ynamics. The ynamics of the three circuits can be written in a common form, m A = F(A) δm A, where F(A) is in one of the following forms: A = κm A γa, (2.23) F unreg (A) = α B, F repress (A) = We choose the parameters to be α B 1 + (A/K) n + α 0, F act (A) = α A(A/K) n 1 + (A/K) n + α B. α A = nm/s, α B = 0.5 nm/s, α 0 = nm/s, κ = s 1, δ = s 1, γ = s 1, K = 10 4 nm, n = 2, corresponing to biologically plausible values. Note that the parameters are chosen so that F(0) α B for each circuit. Figure 2.18a shows the results of simulations comparing the response of the three circuits. We see that initial increase in protein concentration is ientical for each circuit, consistent with our choice of Hill functions an parameters. As the expression level increases, the effects of positive an negative regulation are seen,

40 2.3. TRANSCRIPTIONAL REGULATION 67 Concentration, A (nm) Time (min) (a) Default promoters Concentration, A (nm) unregulate 2000 negative positive Time (min) (b) Moifie promoters Figure 2.18: Simulations for autoregulate gene expression. (a) Non-ajuste expression levels. (b) Equalize expression levels. leaing to ifferent steay state expression levels. In particular, the negative feeback circuit reaches a lower steay state expression level while the positive feeback circuit settles to a higher value. In some situations, it makes sense to ask whether ifferent circuit topologies have ifferent properties that might lea us to choose one over another. In the case where the circuit is going to be use as part of a more complex pathway, it may make the most sense to compare circuits that prouce the same steay state concentration of the protein A. To o this, we must moify the parameters of the iniviual circuits, which can be one in a number of ifferent ways: we can moify the promoter strengths, egraation rates, or other molecular mechanisms reflecte in the parameters. The steay state expression level for the negative autoregulation case can be ajuste by using a stronger promoter (moele by α B ) or ribosome bining site (moele by κ). The equilibrium point for the negative autoregulation case is given by the solution of the equations m A,e = αk n δ(k n + A n e), A e = κ γ m A,e. These couple equations can be solve for m A,e an A e, but in this case we simply nee to fin values α B an κ that give the same values as the unregulate case. For example, if we equate the mrna levels of the unregulate system with that of the negatively autoregulate system, we have α B δ = 1 ( α B K n ) δ K n + A n + α 0 e = α B = (α B α 0 ) Kn + A n e K n, A e = α Bκ δγ, where A e is the esire equilibrium value (which we choose using the unregulate case as a guie).

41 68 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES A similar calculation can be one for the case of positive autoregulation, in this case ecreasing the promoter parameters α A an α B so that the steay state values match. A simple way to o this is to leave α A unchange an ecrease α B to account for the positive feeback. Solving for α B to give the same mrna levels as the unregulate case yiels A n e α B = α B α A K n + A n. e Figure 2.18b shows simulations of the expression levels over time for the moifie circuits. We see now that the expression levels all reach the same steay state value. The negative autoregulate circuit has the property that it reaches the steay state more quickly, ue to the increase rate of protein expression when A is small (α B > α B). Conversely, the positive autoregulate circuit has a slower rate of expression than the constitutive case, since we have lowere the rate of protein expression when A is small. The initial higher an lower expression rates are compensate for via the autoregulation, resulting in the same expression level in steay state. We have escribe how a Hill function can moel the regulation of a gene by a single transcription factor. However, genes can also be regulate by multiple transcription factors, some of which may be activators an some may be repressors, as in the case of combinatorial promoters. The mrna prouction rate can thus take several forms epening on the roles (activators versus repressors) of the various transcription factors. In general, the prouction rate resulting from a promoter that takes as input transcription factors P i for i {1,..., N} will be enote F(P 1,..., P N ). The ynamics of a transcriptional moule are often well capture by the orinary ifferential equations m Pi = F(P 1,..., P N ) δ Pi m Pi, P i = κ Pi m Pi γ Pi P i. (2.24) For a combinatorial promoter with two input proteins, an activator P a an a repressor P r, in which, for example, the activator cannot bin if the repressor is boun to the promoter, the function F(P a, P r ) can be obtaine by employing the competitive bining in the reuce-orer moels of Section 2.1. In this case, assuming the activator has cooperativity n an the repressor has cooperativity m, we obtain the expression (P a /K a ) n F(P a, P r ) = α 1 + (P a /K a ) n + (P r /K r ) m, (2.25) where K a = (K m,a K,a ) (1/n), K r = (K m,r K,r ) (1/m), in which K,a an K,r are the issociation constants of the activator an repressor, respectively, from the DNA promoter site, while K m,a an K m,r are the issociation constants for the cooperative bining reactions for the activator an repressor, respectively. In these expressions,

42 2.3. TRANSCRIPTIONAL REGULATION 69 A A B B C (a) Circuit iagram C Time (min) (b) Simulation Figure 2.19: The incoherent feeforwar loop (type I). (a) A schematic iagram of the circuit. (b) A simulation of the moel in equation (2.28) withβ A = 0.01 µm/min, γ = 0.01 min 1, β B = 1 µm/min, β C = 100 µm/min, K B = µm,ank A = 1 µm. RNA polymerase oes not explicitly appear as it affects the values of the issociation constants an of α. In the case in which the activator is leaky, that is, some transcription still occurs even when there is no activator, the above expression shoul be moifie to (P a /K a ) n F(P a, P r ) = α 1 + (P a /K a ) n + (P r /K r ) m + α 0, (2.26) where α 0 is the basal transcription rate when no activator is present. If the basal rate can still be represse by the repressor, the above expression shoul be moifie to (see Exercise 2.10) F(P a, P r ) = α(p a /K a ) n + α (P a /K a ) n + (P r /K r ) m. (2.27) Example 2.3 (Incoherent feeforwar loops). Combinatorial promoters with two inputs are often use in systems where a logical an is require. As an example, we illustrate here an incoherent feeforwar loop (type I) [4]. Such a circuit is compose of three transcription factors A, B, an C, in which A irectly activates C an B while B represses C. This is illustrate in Figure 2.19a. This is ifferent from a coherent feeforwar loop in which both A an B activate C. In the incoherent feeforwar loop, if we woul like C to be high only when A is high an B is low ( an gate), we can consier a combinatorial promoter in which the activator A an the repressor B competitively bin to the promoter of C. The resulting Hill function is given by the expression in equation (2.25). Depening on the values of the constants, the expression of C is low unless A is high an B is low. The

43 70 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES resulting ODE moel, neglecting the mrna ynamics, is given by the system A = β A γa, B = β A/K A B 1 + (A/K A ) γb, C = β A/K A C 1 + (A/K A ) + (B/K B ) γc, (2.28) in which we have assume no cooperativity of bining for both the activator an the repressor. If we view β A as an input to the system an C as an output, we can investigate how this output respons to a suen increase of β A. Upon a suen increase of β A, protein A buils up an bins to the promoter of C initiating transcription, so that protein C starts getting prouce. At the same time, protein B is prouce an accumulates until it reaches a large enough value to repress C. Hence, we can expect a pulse of C prouction for suitable parameter values. This is shown in Figure 2.19b. Note that if the prouction rate constant β C is very large, a little amount of A will cause C to immeiately ten to a very high concentration. This explains the large initial slope of the C signal in Figure 2.19b. 2.4 Post-transcriptional regulation In aition to regulation of expression through moifications of the process of transcription, cells can also regulate the prouction an activity of proteins via a collection of other post-transcriptional moifications. These inclue methos of moulating the translation of proteins, as well as affecting the activity of a protein via changes in its conformation. Allosteric moifications to proteins In allosteric regulation, a regulatory molecule, calle allosteric effector, bins to a site separate from the catalytic site (active site) of an enzyme. This bining causes a change in the conformation of the protein, turning off (or turning on) the catalytic site (Figure 2.20). An allosteric effector can either be an activator or an inhibitor, just like inucers work for activation or inhibition of transcription factors. Inhibition can either be competitive or not competitive. In the case of competitive inhibition, the inhibitor competes with the substrate for bining the enzyme, that is, the substrate can bin to the enzyme only if the inhibitor is not boun. In the case of non-competitive inhibition, the substrate can be boun to the enzyme even if the latter is boun to the inhibitor. In this case, however, the prouct may not be able to form or may form at a lower rate, in which case, we have partial inhibition.

44 2.4. POST-TRANSCRIPTIONAL REGULATION 71 Allosteric site Protein Catalytic site Regulatory molecule Substrate Conformational change Protein Figure 2.20: In allosteric regulation, a regulatory molecule bins to a site separate from the catalytic site (active site) of an enzyme. This bining causes a change in the threeimensional conformation of the protein, turning off (or turning on) the catalytic site. Figure aapte from Activation can be absolute or not. Specifically, an allosteric effector is an absolute activator when the enzyme can bin to the substrate only when the enzyme is boun to the allosteric effector. Otherwise, the allosteric effector is a non-absolute activator. In this section, we erive the expressions for the prouction rate of the active protein in an enzymatic reaction in the two most common cases: when we have a (non-competitive) inhibitor I or an (absolute) activator A of the enzyme. Allosteric inhibition Consier the stanar enzymatic reaction E + S a ES k E + P, in which enzyme E bins to substrate S an transforms it into the prouct P. Let I be a (non-competitive) inhibitor of enzyme E so that when E is boun to I, the complex EI can still bin to substrate S, however, the complex EIS is non-prouctive, that is, it oes not prouce P. Then, we have the following aitional reactions: E + I k + EI, ES + I k + EIS, EI + S a k k EIS, in which, for simplicity of notation, we have assume that the issociation constant between E an I oes not epen on whether E is boun to the substrate S. Similarly, we have assume that the issociation constant of E from S oes not epen

45 72 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES on whether the inhibitor I is boun to E. Aitionally, we have the conservation laws: E tot = E + [ES] + [EI] + [EIS], S tot = S + P + [ES] + [EIS]. The prouction rate of P is given by P/ = k[es]. Since bining reactions are very fast, we can assume that all the complexes concentrations are at the quasisteay state. This gives [EIS] = a [EI] S, [EI] = k + k E I, [ES] = S E K m, where K m = ( + k)/a is the Michaelis-Menten constant. Using these expressions, the conservation law for the enzyme, an the fact that a/ 1/K m, we obtain E = E tot (I/K + 1)(1 + S/K m ), with K = k /k +, so that an, as a consequence, S E tot [ES] = S + K m 1 + I/K ( )( ) P = ke 1 S tot. 1 + I/K S + K m In our earlier erivations of the Michaelis-Menten kinetics V max = ke tot was calle the maximal velocity, which occurs when the enzyme is completely saturate by the substrate (Section 2.1, equation (2.9)). Hence, the effect of a non-competitive inhibitor is to ecrease the maximal velocity V max to V max /(1 + I/K ). Another type of inhibition occurs when the inhibitor is competitive, that is, when I is boun to E, the complex EI cannot bin to protein S. Since E can either bin to I or S (not both), I competes against S for bining to E (see Exercise 2.13). Allosteric activation In this case, the enzyme E can transform S to its active form only when it is boun to A. Also, we assume that E cannot bin S unless E is boun to A (from here, the name absolute activator). The reactions shoul be moifie to E + A k + EA, EA + S a k EAS k P + EA, with conservation laws E tot = E + [EA] + [EAS], S tot = S + P + [EAS].

46 2.4. POST-TRANSCRIPTIONAL REGULATION 73 Maximal velocity K = 0.5 K = 1 K = 1.5 Maximal velocity Inhibitor concentration, I (a) Allosteric inhibitor Activator concentration, A (b) Allosteric activator Figure 2.21: Maximal velocity in the presence of allosteric effectors (inhibitors or activators). The plots in (a) show the maximal velocity V max /(1 + I/K )asafunctionofthe inhibitor concentration I. Theplotsin(b)showthemaximalvelocityV max A/(A + K )asa function of the activator concentration A. Theifferent plots show the effect of the issociation constant for V max = 1. The prouction rate of P is given by P/ = k [EAS]. Assuming as above that the complexes are at the quasi-steay state, we have that [EA] = E A K, [EAS] = S [EA] K m, which, using the conservation law for E, leas to E = E tot (1 + S/K m )(1 + A/K ) ( )( ) A S an [EAS] = E tot. A + K S + K m Hence, we have that ( )( ) P = ke A S tot. A + K S + K m The effect of an absolute activator is to moulate the maximal spee of moification by a factor A/(A + K ). Figure 2.21 shows the behavior of the maximal velocity as a function of the allosteric effector concentration. As the issociation constant ecreases, that is, the affinity of the effector increases, a very small amount of effector will cause the maximal velocity to reach V max in the case of the activator an 0 in the case of the inhibitor. Another type of activation occurs when the activator is not absolute, that is, when E can bin to S irectly, but cannot activate S unless the complex ES first bins A (see Exercise 2.14).

47 74 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Covalent moifications to proteins In aition to regulation that controls transcription of DNA into mrna, a variety of mechanisms are available for controlling expression after mrna is prouce. These inclue control of splicing an transport from the nucleus (in eukaryotes), the use of various seconary structure patterns in mrna that can interfere with ribosomal bining or cleave the mrna into multiple pieces, an targete egraation of mrna. Once the polypeptie chain is forme, aitional mechanisms are available that regulate the foling of the protein as well as its shape an activity level. One of the most common types of post-transcriptional regulation is through the covalent moification of proteins, such as through the process of phosphorylation. Phosphorylation is an enzymatic process in which a phosphate group is ae to a protein an the resulting conformation of the protein changes, usually from an inactive configuration to an active one. The enzyme that as the phosphate group is calle a kinase an it operates by transferring a phosphate group from a boun ATP molecule to the protein, leaving behin ADP an the phosphorylate protein. Dephosphorylation is a complementary enzymatic process that can remove a phosphate group from a protein. The enzyme that performs ephosphorylation is calle a phosphatase. Figure 2.3 shows the process of phosphorylation in more etail. Since phosphorylation an ephosphorylation can occur much more quickly than protein prouction an egraation, it is use in biological circuits in which a rapi response is require. One common pattern is that a signaling protein will bin to a ligan an the resulting allosteric change allows the signaling protein to serve as a kinase. The newly active kinase then phosphorylates a secon protein, which moulates other functions in the cell. Phosphorylation cascaes can also be use to amplify the effect of the original signal; we will escribe this in more etail in Section 2.5. Kinases in cells are usually very specific to a given protein, allowing etaile signaling networks to be constructe. Phosphatases, on the other han, are much less specific, an a given phosphatase species may ephosphorylate many ifferent types of proteins. The combine action of kinases an phosphatases is important in signaling since the only way to eactivate a phosphorylate protein is by removing the phosphate group. Thus phosphatases are constantly turning off proteins, an the protein is activate only when sufficient kinase activity is present. Phosphorylation of a protein occurs by the aition of a charge phosphate (PO 4 ) group to the serine (Ser), threonine (Thr) or tyrosine (Tyr) amino acis. Similar covalent moifications can occur by the attachment of other chemical groups to select amino acis. Methylation occurs when a methyl group (CH 3 ) is ae to lysine (Lys) an is use for moulation of receptor activity an in moifying histones that are use in chromatin structures. Acetylation occurs when an acetyl group (COCH 3 ) is ae to lysine an is also use to moify histones. Ubiquitina-

48 2.4. POST-TRANSCRIPTIONAL REGULATION 75 Kinase Z ATP Z ADP Z Input C 1 X X * X Substrate X P Y Output (a) General iagram C 2 Y + P Y Phosphatase (b) Detaile view Figure 2.22: (a) General iagram representing a covalent moification cycle. (b) Detaile view of a phosphorylation cycle incluing ATP, ADP, an the exchange of the phosphate group p. tion refers to the aition of a small protein, ubiquitin, to lysine; the aition of a polyubiquitin chain to a protein targets it for egraation. Here, we focus on reversible cycles of moification, in which a protein is interconverte between two forms that iffer in activity. At a high level, a covalent moification cycle involves a target protein X, an enzyme Z for moifying it, an a secon enzyme Y for reversing the moification (see Figure 2.22). We call X the activate protein. There are often allosteric effectors or further covalent moification systems that regulate the activity of the moifying enzymes, but we o not consier this ae level of complexity here. The reactions escribing this system are given by the following two enzymatic reactions, also calle a two-step reaction moel: Z + X a 1 k 1 C 1 X + Z, Y + X a 2 k 2 C 2 X + Y, 1 2 in which we have let C 1 be the kinase/protein complex an C 2 be the active protein/phosphatase complex. The corresponing ifferential equation moel is given by Z = a 1Z X + (k )C 1, X = a 1Z X + 1 C 1 + k 2 C 2, C 1 = a 1 Z X ( 1 + k 1 )C 1, X = k 1 C 1 a 2 Y X + 2 C 2, C 2 = a 2 Y X ( 2 + k 2 )C 2, Y = a 2Y X + ( 2 + k 2 )C 2. Furthermore, we have that the total amounts of enzymes Z an Y are conserve. Denote the total concentrations of Z, Y, an X by Z tot, Y tot, an X tot, respectively.

49 76 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Then, we have also the conservation laws Z +C 1 = Z tot, Y +C 2 = Y tot, X + X +C 1 +C 2 = X tot. Using the first two conservation laws, we can reuce the above system of ifferential equations to the following one: C 1 = a 1 (Z tot C 1 ) X ( 1 + k 1 )C 1, X = k 1 C 1 a 2 (Y tot C 2 ) X + 2 C 2, C 2 = a 2 (Y tot C 2 ) X ( 2 + k 2 )C 2. As in the case of the enzymatic reaction, this system cannot be analytically integrate. To simplify it, we can perform a similar approximation as one for the enzymatic reaction. In particular, the complexes concentrations C 1 an C 2 reach their steay state values very quickly uner the assumption a 1 Z tot, a 2 Y tot, 1, 2 k 1, k 2. Therefore, we can approximate the above system by substituting for C 1 an C 2 their steay state values, given by the solutions to an a 1 (Z tot C 1 ) X ( 1 + k 1 )C 1 = 0 a 2 (Y tot C 2 ) X ( 2 + k 2 )C 2 = 0. By solving these equations, we obtain that an C 2 = C 1 = Y totx X + K m,2, with K m,2 = 2 + k 2 a 2, Z totx X + K m,1, with K m,1 = 1 + k 1 a 1. As a consequence, the moel of the phosphorylation system can be well approximate by X Z tot X Y tot K m,2 = k 1 a 2 X + K m,1 X X Y tot X K m,2 X, + K m,2 which, consiering that a 2 K m,2 2 = k 2, leas finally to X Z tot X Y tot X = k 1 k 2 X + K m,1 X. (2.29) + K m,2 We will come back to the moeling of this system after we have introuce singular perturbation theory, through which we will be able to perform a formal analysis an mathematically characterize the assumptions neee for approximating the

50 2.4. POST-TRANSCRIPTIONAL REGULATION 77 original system by the first-orer moel (2.29). Also, note that X shoul be replace by using the conservation law by X = X tot X C 1 C 2, which can be solve for X using the expressions of C 1 an C 2. Uner the common assumption that the amount of enzyme is much smaller than the amount of substrate (Z tot,y tot X tot )[36], we have that X X tot X [36], leaing to a form of the ifferential equation (2.29) that is simple enough to be analyze mathematically. Simpler moels of phosphorylation cycles can be consiere, which oftentimes are instructive as a first step to stuy a specific question of interest. In particular, the one-step reaction moel neglects the complex formation in the two enzymatic reactions an simply moels them as a single irreversible reaction (see Exercise 2.12). It is important to note that the spee of enzymatic reactions, such as phosphorylation an ephosphorylation, is usually much faster than the spee of protein prouction an protein ecay. In particular, the values of the catalytic rate constants k 1 an k 2, even if changing greatly from organism to organism, are typically several orers of magnitue larger than protein ecay an can be on the orer of 10 3 min 1 in bacteria where typical rates of protein ecay are about 0.01 min 1 ( Ultrasensitivity One relevant aspect of the response of the covalent moification cycle to its input is the sensitivity of the steay state characteristic curve, that is, the map that etermines the equilibrium value of the output X corresponing to a value of the input Z tot. Specifically, which parameters affect the shape of the steay state characteristic is a crucial question. To stuy this, we set X / = 0 in equation (2.29). Using the approximation X X tot X, efining K 1 := K m,1 /X tot an K 2 := K m,2 /X tot, we obtain ( y := k 1Z tot = X /X tot K 1 + (1 X /X tot ) ) k 2 Y tot ( K 2 + X /X tot )(1 X /X tot ). (2.30) Since y is proportional to the input Z tot, we stuy the equilibrium value of X as a function of y. This function is usually characterize by two key parameters: the response coefficient, enote R, an the point of half maximal inuction, enote y 50.Lety α enote the value of y corresponing to having X equal α% of the maximum value of X obtaine for y =, which is equal to X tot. Then, the response coefficient is efine as R := y 90 y 10, an measures how switch-like the response of X is to changes in y (Figure 2.23). When R 1 the response becomes switch-like. In the case in which the steay state characteristic is a Hill function, we have that X = (y/k) n /(1 + (y/k) n ), so

51 78 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES X * /X tot 0.5 R = y 90 y y 10 y 90 y Figure 2.23: Steay state characteristic curve showing the relevance of the response coefficient for ultrasensitivity. As R 1, the points y 10 an y 90 ten to each other. that y α = (α/(100 α)) (1/n) an as a consequence R = (81) (1/n), or equivalently n = log(81) log(r). Hence, when n = 1, that is, the characteristic is of the Michaelis-Menten type, we have that R = 81, while when n increases, R ecreases. Usually, when n > 1 the response is referre to as ultrasensitive an the formula n = log(81)/log(r) is often employe to estimate the apparent Hill coefficient of an experimentally obtaine steay state characteristic since R can be calculate irectly from the ata points. In the case of the current system, from equation (2.30), we have that so that y 90 = ( K ) 0.9 ( K ) 0.1 an y 10 = ( K ) 0.1 ( K ) 0.9, R = 81 ( K )( K ) ( K )( K ). (2.31) As a consequence, when K 1, K 2 1, we have that R 81, which gives a Michaelis- Menten type of response. If instea K 1, K 2 0.1, we have that R 1, which correspons to a theoretical Hill coefficient n 1, that is, a switch-like response (Figure 2.24). In particular, if we have, for example, K 1 = K 2 = 10 2, we obtain an apparent Hill coefficient greater than 13. This type of ultrasensitivity is usually referre to as zero-orer ultrasensitivity. The reason for this name is ue to the fact that when K m,1 is much smaller than the total amount of protein substrate X tot, we have that Z tot X/(K m,1 + X) Z tot. Hence, the forwar moification rate is zero orer in the substrate concentration (no free enzyme is left, all is boun to the substrate). One can stuy the behavior also of the point of half maximal inuction y 50 = K K ,

52 2.4. POST-TRANSCRIPTIONAL REGULATION 79 X /Xtot K m,1 = K m,2 = X tot K m,1 = K m,2 = 0.1X tot K m,1 = K m,2 = 0.01X tot y Figure 2.24: Steay state characteristic curve of a covalent moification cycle as afunction of the Michaelis-Menten constants K m,1 an K m,2. to fin that as K 2 increases, y 50 ecreases an that as K 1 increases, y 50 increases. Phosphotransfer systems Phosphotransfer systems are also a common motif in cellular signal transuction. These structures are compose of proteins that can phosphorylate each other. In contrast to kinase-meiate phosphorylation, where the phosphate onor is usually ATP, in phosphotransfer the phosphate group comes from the onor protein itself (Figure 2.25). Each protein carrying a phosphate group can onate it to the next protein in the system through a reversible reaction. Let X be a protein in its inactive form an let X be the same protein once it has been activate by the aition of a phosphate group. Let Z be a phosphate onor, that is, a protein that can transfer its phosphate group to the acceptor X. The stanar phosphotransfer reactions can be moele accoring to the two-step reaction moel Z + X k 1 C 1 k2 k 3 k4 X + Z, in which C 1 is the complex of Z boun to X boun to the phosphate group. Aitionally, we assume that protein Z can be phosphorylate an protein X ephosphorylate by other phosphorylation reactions by which the phosphate group is taken to an remove from the system. These reactions are moele as one-step reactions epening only on the concentrations of Z an X, that is: Z π 1 Z, X π 2 X. Proteins X an Z are conserve in the system, that is, X tot = X + C 1 + X an Z tot = Z + C 1 + Z. We view the total amount of Z, Z tot, as the input to our system an the amount of phosphorylate form of X, X, as the output. We are intereste in the steay state characteristic curve escribing how the steay state value of X epens on the value of Z tot.

53 80 CHAPTER 2. DYNAMIC MODELING OF CORE PROCESSES Input Z Z * Z P Z ADP C X X * X X P Output (a) General iagram (b) Detaile view Figure 2.25: (a) Diagram of a phosphotransfer system. (b) Proteins X an Z are transferring the phosphate group p to each other. The ifferential equation moel corresponing to this system is given by the equations C 1 ( = k 1 Xtot X ) C 1 Z k 3 C 1 k 2 C 1 + k 4 X (Z tot C 1 Z ), Z = π 1 (Z tot C 1 Z ( ) + k 2 C 1 k 1 Xtot X ) C 1 Z, X = k 3 C 1 k 4 X (Z tot C 1 Z ) π 2 X. (2.32) The steay state transfer curve is shown in Figure 2.26 an it is obtaine by simulating system (2.32) an recoring the equilibrium values of X corresponing to ifferent values of Z tot. The transfer curve is linear for a large range of values of Z tot an can be renere fairly close to a linear relationship for values of Z tot X k 1 = k 1 = 0.1 k 1 = Z tot Figure 2.26: Steay state characteristic curve of the phosphotransfer system. Here, we have set k 2 = k 3 = 0.1s 1, k 4 = 0.1nM 1 s 1, π 1 = π 2 = 3.1 s 1,anX tot = 100 nm.