Simulation of Chemical Reactions

Transcription

1 Simulation of Chemical Reactions Cameron Finucane & Alejandro Ribeiro Dept. of Electrical and Systems Engineering University of Pennsylvania November 2, 2015 Stoch. Systems Analysis Simulation of Chemical Reactions 1

2 Predator-Prey model (Lotka-Volterra system) Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 2

3 A simple Predator-Prey model Populations of X prey molecules and Y predator molecules Three possible reactions (events) Prey reproduction: X 2X Prey consumption to generate predator: X +Y 2Y Predator death: Y Each prey reproduces at rate α Population of X preys αx = rate of first reaction Prey individual consumed by predator individual on chance encounter X prey and Y predator βxy = rate of second reaction β = Rate of encounters between prey and predator individuals Each predator dies off at rate γ Population of Y predators γy = rate of third reaction Stoch. Systems Analysis Simulation of Chemical Reactions 3

4 The Lotka-Volterra equations Study population dynamic X (t) and Y (t) as functions of time t Conventional approach: model system as system of differential eqs. Lotka-Volterra (LV) differential equations Change in prey (dx (t)/dt) = Prey generation - Prey consumption Prey is generated when it reproduces rate αx (t) Prey consumed by predators rate βx (t)y (t) dx (t) dt = αx (t) βx (t)y (t) Predator change (dy (t)/dt) = Predator generation - consumption Predator is generated when it consumes prey rate βx (t)y (t) Predator consumed when it dies off rate γy (t) dy (t) dt = βx (t)y (t) γy (t) Stoch. Systems Analysis Simulation of Chemical Reactions 4

5 Solution of the LV equations LV equations are non-linear but can be solved numerically Population Size X (Prey) Y (Predator) Prey reproduction rate α = 1 Predator death rate γ = 0.1 Predator consumption of prey β = 0.1 Initial state X (0) = 4 Y (0) = Boom and bust cycles Time Start with prey reproduction > consumption prey X (t) increases Predator production picks up (proportional to X (t)y (t)) Predator production > death predator Y (t) increases Eventually prey reproduction < consumption prey X (t) decreases Predator production slows down (proportional to X (t)y (t)) Predator production < death predator Y (t) decreases Prey reproduction > consumption (start over) Stoch. Systems Analysis Simulation of Chemical Reactions 5

6 State space diagram State-space diagram plot Y (t) versus X (t) System constrained to single orbit given by initial state X (0), Y (0) Y (Predator) (4,10) (4,6) (4,4) (4,2) (X(0),Y(0)) = (4,1) X (Prey) Buildup: Prey increases fast, predator increases slowly (move right and slightly up) Boom: Predator increases fast depleting prey (move up and left) Bust: When prey is depleted predator collapses (move down almost straight) Stoch. Systems Analysis Simulation of Chemical Reactions 6

7 Two observations Too much regularity for a natural system (exact periodicity forever) 20 X (Prey) Y (Predator) 15 Population Size Time X (t), Y (t) modeled as continuous but actually discrete. Is this a problem? If X (t), Y (t) large can interpret as concentrations (molecules/volume) Accurate in many cases (millions of molecules) If X (t), Y (t) small does not make sense Our simulation had 7/100 prey at some point Population Size X 0.07 X (Prey) Y (Predator) 2 There is an extinction event we are missing Time Stoch. Systems Analysis Simulation of Chemical Reactions 7

8 Things deterministic model explains (or does not) Deterministic model is useful E.g. boom and bust cycles Important property that the model predicts and explains But it does not capture some aspects of the system. E.g., Non-discrete population sizes (unrealistic fractional molecules) No random variation (unrealistic regularity) Possibly missing important phenomena e.g., extinction Shortcomings most pronounced when number of molecules is small Important in biochemistry at cellular level (1 5 molecules typical) Address these shortcomings through a stochastic model Stoch. Systems Analysis Simulation of Chemical Reactions 8

9 Stochastic model Three possible reactions (events) occurring at rates c 1, c 2 and c 3 Prey reproduction: X c 1 2X Prey consumption to generate predator: X +Y c 2 2Y Predator death: Y c 3 Denote as X (t), Y (t) number of molecules by time t Can model X (t), Y (t) as continuous time Markov chains (CTMCs)? Large population size argument not applicable because we want to model systems with small number of molecules Stoch. Systems Analysis Simulation of Chemical Reactions 9

10 Stochastic model (continued) Consider system with 1 prey molecule x and 1 predator molecule y Let T 2 (1, 1) be the time until x reacts with y Since T 2 (1, 1) is the time until x encounters y and x and y move randomly around it is reasonable to model T 2 (1, 1) as memoryless P [ T 2 (1, 1) > s + t T2 (1, 1) > s ] = P [T 2 (1, 1) > t] T 2 (1, 1) is exponential with parameter c 2 If there are X prey and Y predator there are XY possible reactions between a specimen of type X and a specimen of type Y Let T 2 (X, Y ) be the time until the first of these reactions occurs Min. of exponential RVs is exponential with summed parameters T 2 (X, Y ) is exponential with parameter c 2 XY Likewise time T 1 (X ) until first reaction of type 1 is exponential with parameter c 1 X and time T 3 (Y ) is exponential with parameter c 3 Y Stoch. Systems Analysis Simulation of Chemical Reactions 10

11 CTMC model If reaction times are exponential can model as CTMC CTMC state is pair (X, Y ) with nr. of prey and predator molecules X 1, Y +1 c 1(X 1) X, Y +1 c 3(Y + 1) c 2 XY c 1(X 1) X 1, Y c 2X (Y 1) c 3(Y + 1) c 2(X + 1)Y c 1 X X, Y X +1, Y c 2(X + 1)(Y 1) c 3 Y c 3Y Transition rates (X, Y ) (X + 1, Y ): Reaction 1 = c 1X (X, Y ) (X 1, Y +1): Reaction 2 = c 2X (X, Y ) (X, Y 1): Reaction 3 = c 3X X, Y 1 c 1X X +1, Y 1 Stoch. Systems Analysis Simulation of Chemical Reactions 11

12 Simulation of CTMC model Use CTMC model to simulate Predator-prey model Initial conditions are X (0) = 50 prey and Y (0) = 100 predator Population Size X (Prey) Y (Predator) Prey reproduction rate c 1 = 1 reactions/second Rate of predator consumption of prey c 2 = reactions/second Predator death rate c 3 = 0.6 reactions/second Time Boom and bust cycles are still the dominant feature of the system but random variations are apparent Stoch. Systems Analysis Simulation of Chemical Reactions 12

13 CTMC model in state space Plot Y (t) versus X (t) for the CTMC state space representation Y (Predator) X (Prey) There is not a single fixed orbit as before Can think of this orbit as a perturbed version of deterministic orbit Stoch. Systems Analysis Simulation of Chemical Reactions 13

14 Effects of different population sizes Chance of extinction captured by CTMC model (top plots) Population Size X(0) = 8, Y(0) = 16 X (Prey) Y (Predator) Population Size X(0) = 16, Y(0) = 32 X (Prey) Y (Predator) Population Size X(0) = 32, Y(0) = 64 X (Prey) Y (Predator) Time Time Time X(0) = 64, Y(0) = 128 X (Prey) Y (Predator) X(0) = 128, Y(0) = 256 X (Prey) Y (Predator) X(0) = 256, Y(0) = 512 X (Prey) Y (Predator) Population Size Population Size Population Size Time Time (Notice that Y-axis scales are different) Time Stoch. Systems Analysis Simulation of Chemical Reactions 14

15 Conclusions and the road ahead Deterministic vs. stochastic modeling Deterministic modeling is simpler Captures dominant features (boom & bust cycles) Stochastic simulation more complex Less regularity, (all runs are different, state orbit not fixed) Captures effects missed by deterministic solution (extinction) Gillespie s algorithm. Forthcoming Building a CTMC model for every system of reactions is cumbersome Impossible if there are tens or hundreds of types and reactions Gillespie s algorithm is just a general way of writing a simulation code for a generic system of chemical reactions Stoch. Systems Analysis Simulation of Chemical Reactions 15

16 Gillespie s algorithm Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 16

17 Simulation of chemical reactions Chemical system with m reactant types and n possible reactions Reactant quantities change over time as reactions occur Nr. of type j reactants at time t denoted as X j (t) System s state vector X(t) := [X 1 (t), X 2 (t),..., X j (t)] T To specify i-th reaction reactants, products and rates R i : s l i1x 1 + s l i2x s l imx m h i (X) s r i1x 1 + s r i2x s r imx m (si1 l molecules of type 1) (sl im molecules of type m) react to yield (si1 r of type 1) (sr im of type m) Rate of reaction h i (X) depends on number of molecules present Let T i (X) denote the time until the i-th reaction when state is X Stoch. Systems Analysis Simulation of Chemical Reactions 17

18 Stoichiometry matrices Can be more conveniently written using matrices Define vector of rates h(x) = [h 1(X), h 2(X),..., h n(x)] T Define stoichiometry left matrix S (l) with elements s l ij Define stoichiometry right matrix S (r) with elements s r ij Write system of chemical reactions as S (l) X h(x) S (r) X X 1 X 2 = X X 1 X 2 = X s l 11 sl 12 sl 1m s l i1 s l i2 s l im s l n2 sl n2 sl nm X m s l 11 X sl 1m Xm s l i1 X sl im Xm s l n1 X sl nm Xm = s r 11 sr 12 sr 1m s r i1 s r i2 s r im s r n2 sr n2 sr nm X m s r 11 X sr 1m Xm s r i1 X sr im Xm s r n1 X sr nm Xm S (l) S (l) X S (r) S (r) X Stoch. Systems Analysis Simulation of Chemical Reactions 18

19 Example 1: Dimerization kinetics Molecule can exist in simple form P and as a dimer D Define vector X := [P, D] T Possible reactions are dimerization and dissociation R 1 (Dimerization): 2P h1(x) D R 2 (Dissociation): D h2(x) 2P Rates and stoichiometry matrices S (l) and S (r) given by [ ] [ ] [ S (l) 2 0 =, S (r) 0 1 h1 (X) =, h(x) = h 2 (X) Rewrite equations more compactly as S (l) X h(x) S (r) X ] Stoch. Systems Analysis Simulation of Chemical Reactions 19

20 Example 2: Enzymatic reaction Substrate S converted to product P. Enzyme E catalyzes conversion Converting S into P directly requires significant energy Enzyme E reacts with S to form intermediate molecule SE (binding) Molecule SE then separates into product P liberating E (conversion) This cycle requires less energy than direct conversion SE may also separate back into S and E (dissociation) Possible reactions are binding, conversion and dissociation, then R 1 (Binding): R 2 (Dissociation): SE R 3 (Conversion): S + E h1(x) SE SE h 2(X) S + E h 3(X) E + P Stoch. Systems Analysis Simulation of Chemical Reactions 20

21 Example 2: Enzymatic reaction (continued) System state represented by vector X := [S, E, SE, P] T Stoichiometry matrices S (l) and S (r) given by S E SE P S E SE P R S (l) = R 2 S (r) = R Reaction rate vector h(x) = [h 1 (X), h 2 (X), h 3 (X)] T Rewrite equations more compactly as S (l) X h(x) S (r) X R 1 R 2 R 3 Stoch. Systems Analysis Simulation of Chemical Reactions 21

22 Second order reaction Consider second order reaction R i : X 1 + X 2... (two reactants) Let T i (X 1, X 2 ) be time until R occurs when there are X 1 type 1 and X 2 type 2 molecules Have seen that T i (X 1, X 2 ) is exponentially distributed with rate h i (X) = h i (X 1, X 2 ) = c i X 1 X 2 Constant c i measures reactivity of X 1 and X 2 Argument T i (1, 1) memoryless (depends on chance encounter) Thus T i (1, 1) is exponential with, say, parameter c i T i (X 1, X 2 ) is the minimum of X 1 X 2 exponentials T i (X 1, X 2 ) exponential with parameter c i X 1 X 2 Stoch. Systems Analysis Simulation of Chemical Reactions 22

23 Second order involving molecules of same type Second order reaction with two molecules of same type R i : X 1 + X 1... Hazard depends on the number of molecules X 1, i.e. h i (X) = h i (X 1 ) Reaction does not occur if there is a single molecule If there are 2 molecules T i (2) is exponential with parameter, say, c i For arbitrary X 1 there are X 1 (X 1 1)/2 possible encounters Then, T i (X 1 ) is exponential with parameter h i (X) = h i (X 1 ) = c i X 1 (X 1 1)/2 c i X 1 (X 1 1)/2 substantially different from c i X 2 1 /2 for small X 1 Stoch. Systems Analysis Simulation of Chemical Reactions 23

24 Zero-th, first and higher order reactions Zero-th order reaction R i : X 1 (spontaneous generation) Assume an exponential model with constant rate h i = c i Used to model exogenous factors (and biblical phenomena) First order reaction R i : X 1... (decay) Exponential with rate h i (X) = h i (X 1 ) = c i X 1 Higher order reactions involving more than two reactants E.g., third order reaction R i : X 1 + X 2 + X 3 X 4 Time until next R i reaction exponential. Hazard: h i (X) = c i X 1 X 2 X 3 Reactions of order more than 2 are rare Most likely, R i is encapsulating two second order reactions X 1 + X 2 X 5, X 5 + X 3 X 4 Stoch. Systems Analysis Simulation of Chemical Reactions 24

25 The hazard function All reaction times are exponential RVs CTMC with state X Hazards h i (X) determine transition rates of CTMC Hazards for zero-th, first and second order reactions (for reference) Order Reaction Rate zero-th c c first X 1 c cx1 second X 1 + X 2 c cx1 X 2 second 2X 1 c cx1 (X 1 1)/2 Probability of reaction R i happening in infinitesimal time ɛ is That s why the name hazard P [T i (X) < ɛ] = h i (X)ɛ + o(ɛ) Stoch. Systems Analysis Simulation of Chemical Reactions 25

26 State transition for given reaction State is X(t) = X. Reaction R i occurs. Next state X(t + dt) = Y? Number of reactants per type = = i-th row of left stoichiometry matrix s (l) i = [s l i1, sl i2,..., sl im ]T s l i1x 1 + s l i2x s l imx m h i (X)... Number of products per type = = i-th row of right stoichiometry matrix s (r) i = [s r i1, sr i2,..., sr im ]T... h i (X) s r i1x 1 + s r i2x s r imx m X decreases by nr. of reactants and increases by nr. of products Next sate is Y = X s (l) i + s (r) i (upon reaction R i ) Stoch. Systems Analysis Simulation of Chemical Reactions 26

27 Transition rates and probabilities q(x, Y) = transition rate from state X to state Y. Given by ( ) q X, X s (l) i + s (r) i = h i (X), i = 1,..., n Transition from state X to X s (l) i + s (r) i when reaction R i occurs ν(x) = Transition rate out of X into any state (any reaction occurs) ν(x) = n i=1 ( q X, X s (l) i ) + s (r) i = n h i (X) i=1 P(X, Y) = Prob. of going into Y given transition out of X occurs ( ) ( ) q X, X s (l) P X, X s (l) i + s (r) i + s (r) i i = = h i(x) ν(x) ν(x) Probability that i-th reaction occurs given that a reaction occurred Stoch. Systems Analysis Simulation of Chemical Reactions 27

28 Gillespie s algorithm Gillespie s algorithm = Simulation of CTMC Input: Stoichiometry matrices S (l) and S (r). Initial state X(0) Output: Molecule numbers as a function of time X(t) (1) Initialize time and CTMC s state t = 0, X = X(0) (2) Calculate all hazards h i (X) (3) Calculate transition rate ν(x) = n i=1 h i(x) (4) Draw random time of next reaction t Exp ( ν(x) ) (5) Advance time to t = t + t (6) Draw reaction at time t + t R i drawn with prob. h i (X)/ν(X) (7) Update state vector to account for this reaction X s (l) i (8) Repeat from (2) + s (r) i Stoch. Systems Analysis Simulation of Chemical Reactions 28

29 Dimerization Kinetics Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 29

30 Dimerization Dimerization occurs when two like molecules join together Many proteins (P) will form dimers (D) Dimerization may be rare in relative terms, but significant in absolute terms at high concentration. For this reason plays important role in auto-regulation of protein production Possible reactions are dimerization and dissociation R 1 (Dimerization): 2P c 1 D R 2 (Dissociation): D c 2 2P Dimerization rare and dimers unstable c 2 c 1 Stoichiometry matrices S (l) and S (r) given by [ ] [ S (l) 2 0 =, S (r) 0 1 = Rate of reaction 1 is h 1(X) = c 1P(P 1)/2. Reaction 2 is h 2(X) = c 2D ], Stoch. Systems Analysis Simulation of Chemical Reactions 30

31 Gillespie s algorithm for dimerization kinetics (1) Initialize time and CTMC s state t = 0, P = P(0), D = D(0) (2) Calculate hazards h 1 (X) = c 1 P(P 1)/2, h 2 (X) = c 2 D (3) Calculate transition rate ν(x) = c 1 P(P 1)/2 + c 2 D (4) Draw random time of next reaction t exp ( ν(x) ) = exp ( c 1 P(P 1)/2 + c 2 D ) (5) Advance time to t = t + t (6) Draw reaction at time t + t P [Dimerization:] = c 1 P(P 1)/2/ν(X) P [Dissociation:] = c 2 D/ν(X) (7) Update state vector Dimerization: P = P 2, D = D + 1 Dissociation: P = P + 2, D = D 1 (8) Repeat from (2) Stoch. Systems Analysis Simulation of Chemical Reactions 31

32 Stochastic simulation of dimerization kinetics Run of Gillespie s algorithm for dimerization kinetics Initial condition P(0) = 301, D(0) = 0 (protein only) # of molecules Time [P] [P 2 ] Dimerization hazard c 1 = reactions sec./molecule 2 Dissociation hazards c 2 = reactions sec./molecule c = [c 1, c 2 ] T = [ , 0.2] T P and D stabilize at point where dimerization and dissociation become equally likely Stoch. Systems Analysis Simulation of Chemical Reactions 32

33 Information that can be obtained from simulations E.g., consider nr. of protein molecules P (P(t) + 2D(t) is constant) Mean and standard deviation of P versus time? Right graph mean and ±3(standard deviations) over 10 4 trials Left graph shows 20 trials Vary around mean path but stay within ±3-standard deviations 300 Mean ±3! 300 [P] # of molecules 150 # of molecules Time Time Stoch. Systems Analysis Simulation of Chemical Reactions 33

34 Steady-state probability distribution Time t = 10 seconds approximate PMF over 10 4 trials Can use ergodicity instead Probability P(t=10) Bell-shaped. Only odd values of P are possible Runs are all odd or all even depending on initial condition Stoch. Systems Analysis Simulation of Chemical Reactions 34

35 Enzymatic reactions Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 35

36 Enzymes Substrate S converted into product P by action of enzyme E Intermediate product SE generated by combination of E and S SE later separates into product P liberating the enzyme E SE may also dissociate into S and E Enzymes can act as catalysts for reactions that would otherwise rarely or never take place Possible reactions are binding, dissociation and conversion R 1 (Binding): R 2 (Dissociation): SE R 3 (Conversion): S + E c1 SE SE c 2 S + E c 3 P + E Dissociation typically not significant because c 2 c 3 Stoch. Systems Analysis Simulation of Chemical Reactions 36

37 Enzymatic reactions (continued) Stoichiometry matrices S (l) and S (r) given by S E SE P S E SE P R S (l) = R 2 S (r) = R Reaction rates are Reaction R 1 (Binding): h 1 (X) = c 1 S E, Reaction R 2 (Dissociation): h 2 (X) = c 2 SE Reaction R 3 (Conversion): h 3 (X) = c 3 SE R 1 R 2 R 3 Stoch. Systems Analysis Simulation of Chemical Reactions 37

38 Gillespie s algorithm for enzymatic reactions (1) Initialization: t = 0, S = S(0), E = E(0), SE = SE(0), P = P(0) (2) Calculate hazards h 1(X) = c 1S E, h 2(X) = c 2SE h 3(X) = c 3SE (3) Calculate transition rate ν(x) = c 1S E + c 2SE + c 3SE (4) Draw random time of next reaction t exp ( ν(x) ) = exp ( c 1S E + c 2SE + c 3SE ) (5) Advance time to t = t + t (6) Draw reaction at time t + t P [Binding:] = c 1S E/ν(X) P [Dissociation:] = c 2SE/ν(X) P [Conversion:] = c 3SE/ν(X) (7) Update state vector Binding: S = S 1, E = E 1, SE = SE + 1 Dissociation: S = S + 1, E = E + 1, SE = SE 1 Conversion: P = P + 1, E = E + 1, SE = SE 1 (8) Repeat from (2) Stoch. Systems Analysis Simulation of Chemical Reactions 38

39 Stochastic simulation of enzymatic reactions Run of Gillespie s algorithm for enzymatic reactions Initialize with only substrate and enzyme present S(0) = 301, E(0) = 120, SE(0) = 0, P(0) = 0 # of molecules Time [S] [E] [SE] [P] Binding hazard c 1 = reactions sec./molecule 2 Dissociation hazard c 2 = 10 4 reactions sec./molecule Conversion hazard reactions c 3 = 0.1 sec./molecule c = [c 1, c 2, c 3] T = [ , 10 4, 0.1] T Stoch. Systems Analysis Simulation of Chemical Reactions 39

40 Stochastic simulation (continued) At the beginning substrate and enzyme numbers decline as they bind to each other to form intermediate product SE Intermediate product separates into final product P liberating enzyme E By t = 50 seconds substrate is completely converted into product and enzymes are free. There is no intermediate product either [S] [E] [SE] [P] 200 # of molecules Time Stoch. Systems Analysis Simulation of Chemical Reactions 40

41 Auto-regulatory gene network Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 41

42 Auto-regulation of protein production Simplified model of protein production in prokaryotes Instructions for creating protein (P) encoded in gene (G) To produce protein, gene G is first transcribed into mrna (R) This mrna is passed on to a ribosome to assemble the protein Protein production usually triggered by external stimuli How is it halted? Negative feedback loops called auto-regulatory networks As protein numbers increase, so does presence of a byproduct, E.g., a protein dimer (D) Byproducts show affinity to bind to the gene blocking transcription Halting transcription slows/halts protein production Stoch. Systems Analysis Simulation of Chemical Reactions 42

43 Reactions of the auto-regulatory network Protein production consists of transcription and assembly Transcription: G c 1 G + R Assembly: R c 2 R + P Dimer is generated as a byproduct of protein production Dimerization: 2P c 3 D Dissociation: D c 4 2P Dimer binds to mrna blocking transcription. Blocked gene may be liberated Repression: G + D c 5 GD Liberation: GD c 6 G + D Protein and mrna eventually degrade (mrna degradation common) mrna degradation: R c 7 Protein degradation: P c 8 Stoch. Systems Analysis Simulation of Chemical Reactions 43

44 Approach to modeling We will use rate constants c 1 (transcription) = 0.01 c 2 (assembly) = 10 c 3 (dimerisation) = 1 c 4 (dissociation) = 1 C = c 5 (repression) = 1 c 6 (reverse repression) = 10 c 7 (mrna degradation) = 0.1 c 8 (protein degradation) = 0.01 gene repressed gene G(0)= 10, P 2 G(0) = mrna R(0)= protein P(0)= P dimer 2 (0)= 0 Because of the very small numbers of molecules involved, a continuous deterministic approach would not provide accurate results. Stoch. Systems Analysis Simulation of Chemical Reactions 44

45 The stochastic solution Stochastic simulation. Protein dimer and mrna numbers shown mrna numbers are very small (0, 1 or 2) 2 mrna (R) Protein (P) Protein Dimer (P2) Time Increase in protein & dimer triggered by mrna transcription events Transcription events spread out when protein nrs. are large Transcription events occur more rapidly when there is less protein Stoch. Systems Analysis Simulation of Chemical Reactions 45

46 Notes Because there are a very small number of genes, the stochastic nature of the number of mrna molecules transcribed is very clearly evident. Even though there are larger numbers of P 2, their numbers are affected directly by the mrna transcription events, so stochasticity still dominates. Stoch. Systems Analysis Simulation of Chemical Reactions 46

47 Observing the distribution At steady-state, we find the following PMF for the number of protein molecules (over 10,000 trials, using the property of ergodicity): Probability # of Protein (P) Molecules Notice that the distribution is very evenly centered around 25, showing successful auto-regulation. Stoch. Systems Analysis Simulation of Chemical Reactions 47

48 Lactose digestion (lac operon) Predator-Prey model (Lotka-Volterra system) Gillespie s algorithm Dimerization Kinetics Enzymatic Reactions Auto-regulatory gene network Lactose digestion (lac operon) Stoch. Systems Analysis Simulation of Chemical Reactions 48

49 Auto-regulation of protein production Simplified model of protein production in prokaryotes Instructions for creating proteins encoded in genes To produce proteins, genes are first transcribed into mrna This mrna is passed on to a ribosome to assemble the protein Protein production not immutable. How does it changes over time? Auto regulatory gene networks Production triggered by external stimuli Halted by negative feedback loops through protein byproducts E.g. Production of β-galactosidase to digest glucose Lac-operon (lac for lactose, operon=set of interacting genes) Stoch. Systems Analysis Simulation of Chemical Reactions 49

50 Glucose, Lactose and β-galactosidase Glucose (G) and lactose (L) are variations of sugars Cells use glucose for energy but can reduce lactose to glucose Lactose reduced to glucose by enzyme β-galactosidase (βg) Lactose digestion: L + βg c 1 G + βg Glucose consumption: G c 2 Did not model enzymatic reaction (compare with earlier example) Rate of lactose digestion c 1 L (βg). Glucose consumption c 2 G Producing β-galactosidase is not always necessary Production necessary only when lactose is present and glucose is not Stoch. Systems Analysis Simulation of Chemical Reactions 50

51 Lac-operon, normal state Lac-operon consists of three adjacent genes Promoter, operator and β-galactosidase code (three types in fact) Lac-operon has three possible states, regular, activated and repressed In normal state (Op) transcription proceeds at a small rate c 3 The promoter is a binding place for RNA polymerase (RNAP) RNAP binds to promoter to initiate gene transcription into mrna RNAP mrna promoter operator lac x lac y lac z Model reaction as Regular transcription: Op c 3 Op + mrna Stoch. Systems Analysis Simulation of Chemical Reactions 51

52 Lac-operon in activated state Operon activated (AOp) by catabolite activator protein (CAP) CAP binds upstream of the promoter altering DNA s geometry Thereby facilitating (promoting) binding of RNAP to promoter Hence yielding a faster rate of transcription c 4 c 3 CAP RNAP mrna promoter operator lac x lac y lac z Model reaction as Activated transcription: AOp c 4 AOp + mrna Stoch. Systems Analysis Simulation of Chemical Reactions 52

53 Lac-operon in repressed state Operon repressed (ROp) by lactose repressor protein protein (LRP) LRP encoded by gene adjacent to lac operon, is always expressed and has great affinity with the operator If LRP binds to operator it interferes with RNAP promoter binding Without RNAP, there is no (or minimal) transcription Hence yielding a very slow rate of transcription c 5 c 3 c 4 RNAP L R P mrna promoter operator lac x lac y lac z Model reaction as Repressed transcription: ROp c 5 ROp + mrna Stoch. Systems Analysis Simulation of Chemical Reactions 53

54 Repression control If there is no lactose (L) present lac operon is in repressed state When lactose is present it combines with LRP Thereby preventing repression of lac operon. Lac operon in regular state Small (but not minimal) rate of β-galactosidase production RNAP mrna promoter operator lac x lac y lac z LRP Lactose We model this with the following reactions Operon repression: LRP + Op c 6 ROp Operon liberation: ROp Repressor neutralization: LRP + L Repressor dissociation: LRPL c 7 LRP + Op c 8 LRPL c 9 LRP + L Stoch. Systems Analysis Simulation of Chemical Reactions 54

55 Activation control Prevalence of CAP inversely proportional to glucose levels This involves a complex set of reactions in itself For a preliminary model the following reactions suffice Operon activation: CAP + Op c 10 AOp Operon deactivation: AOp c 11 CAP + Op CAP neutralization: CAP + G c 12 CAPG CAP dissociation: CAPG If glucose is present, CAP is bound to glucose Thereby preventing activation of lac operon Small rate of β-galactosidase production c 13 CAP + G RNAP mrna promoter operator lac x lac y lac z CAP Glucose Stoch. Systems Analysis Simulation of Chemical Reactions 55

56 Glucose, lactose and lac-operon states High lactose and high glucose (glucose preferred) CAP bound to glucose and LRP bound to lactose Operon in regular state, low production of β-galactosidase High lactose and low glucose (lactose only option) CAP bound upstream of promoter and LRP bound to lactose Operon in activated state, high production of β-galactosidase High glucose and low lactose (glucose dominant and preferred) CAP bound to glucose and LRP bound to operator Operon in repressed state, minimal production of β-galactosidase Low glucose and low lactose (no energy source available) CAP bound upstream of promoter and LRP bound to operator Repression dominates, minimal production of β-galactosidase β-galactosidase produced in significant quantities only with high lactose and low glucose concentrations Stoch. Systems Analysis Simulation of Chemical Reactions 56

57 β-galactosidase assembly and decays To complete model we add reactions to account for Assembly of β-galactosidase (βg) enzyme mrna and βg decay Protein synthesis: mrna c 14 mrna + βg mrna decay: mrna c 15 βgalactosidase decay: βg c 16 Stoch. Systems Analysis Simulation of Chemical Reactions 57

58 Reactions modeling digestion of lactose Model of auto-regulatory gene network for digestion of lactose Rates in reactions/minute/molecule or reactions/minute/molecule 2 Lactose digestion: Glucose consumption: Regular transcription: L + βg G Op Activated transcription: AOp Repressed transcription: ROp c 1 G + βg c1 = 1 c 2 c2 = 0.1 c 3 Op + mrna c3 = 0.01 c 4 AOp + mrna c4 = 0.1 c 5 ROp + mrna c5 = Operon repression: LRP + Op c 6 ROp c 6 = 1 Operon liberation: ROp c 7 LRP + Op c7 = 1 Compare rates c 3 -c 5 for lac operon in different states Stoch. Systems Analysis Simulation of Chemical Reactions 58

59 Reactions modeling digestion of lactose (continued) Model of auto-regulatory gene network for digestion of lactose Rates in reactions/minute/molecule or reactions/minute/molecule 2 Repressor neutralization: LRP + L c 8 LRPL c8 = 10 Repressor dissociation: LRPL c 9 LRP + L c9 = 1 Operon activation: CAP + Op c 10 AOp c 10 = 1 Operon deactivation: AOp c 11 CAP + Op c11 = 1 CAP neutralization: CAP + G c 12 CAPG c 12 = 10 CAP dissociation: Protein synthesis: mrna decay: βgalactosidase decay: CAPG mrna mrna βg c 13 CAP + G c13 = 1 c 14 mrna + βgc14 = 1 c 15 c15 = 1 c 16 c16 = 0.1 Notice that LRP and CAP neutralization are fast (rates c 8 and c 12 ) Stoch. Systems Analysis Simulation of Chemical Reactions 59

60 Stochastic simulation: diauxie pattern Initial state L = 50, G = 50, CAP = 10, LRP = 10 Only 1 operon in regular state L G Sugars (glucose and lactose) consumed sequentially Glucose is consumed first After glucose is depleted, lactose converted to glucose After conversion, newly generated glucose is also consumed Yields two growth spurts = diauxie pattern Stoch. Systems Analysis Simulation of Chemical Reactions 60

61 Operon state and diauxie pattern Conversion occurs with operon in activated state L G Repressed Regular Activated Stoch. Systems Analysis Simulation of Chemical Reactions 61

62 mrna transcription & β-galactosidase synthesis Operon activation mrna transcription β-galactosidase synthesis lactose digestion Repressed Regular Activated mrna betag L G Stoch. Systems Analysis Simulation of Chemical Reactions 62