An introduction to mathematical modeling. of signal transduction and gene control networks

Transcription

1 An introduction to mathematical modeling of signal transduction and gene control networks Hans G. Othmer Department of Mathematics University of Minnesota Minneapolis, MN

2 Overview First Lecture: An introduction to mathematical modeling of signal transduction and gene control networks Examples of signal transduction, metabolic and gene control networks What is it we want to understand? The mathematical description of chemical reactions Analytical and computational techniques Second lecture: Analysis of a model of signal transduction and motor control Signal transduction in bacteria The basic input-output behavior Response to steps and slow ramps The gain in the signal transduction pathway The bacterial motor Monte Carlo results

3 Third lecture: Analysis of a model for Drosophila melanogaster segment polarity genes Background on Drosophila development The basic facts about segment polarity genes Description of the Boolean representation Analysis of wild-type and heat-shock behavior Expression patterns in mutants A two-step model

4 Biochemical networks Signal transduction networks: The pathways and the molecular components, such as kinases, G-proteins, sencond messengers,..., involved in transducing a signal from one location to another. Frequently used in the context of transduction of extra- into intracellular signals. Metabolic networks: The pathways and the molecular components (metabolites, enzymes, control factors) involved in the biosynthesis of new components, the conversion of molecular foodstuffs into energy, etc. One of the most important examples is the glycolytic pathway, which converts sugars into energy-storing molecules such as ATP. Gene expression networks: The pathways and components, such as genes, polymerases, transcription factors, etc., that are involved in gene expression, mrna translation, etc.

5 An early view of signal transduction To understand, next, how external objects that strike the sense organs can incite [the machine] to move its members in a thousand different ways: think that (a) the filaments (I have already often told you that these come from the innermost part of the brain and compose the marrow of the nerves) are so arranged that they can very easily be moved by the objects of that sense and that (b) when they are moved, with however little force, they simultaneously pull the parts of the brain from which they come, and by this means open the entrances to certain pores in the internal surface of the brain.. Thus if fire A is near foot B, the particles of this fire have force enough to displace the area of skin they touch; and thus pulling the little thread (cc) which you see attached there, they simultaneously open the entrance to the pore (de) where this thread terminates; just as, pulling on one end of a cord, one simultaneously rings a bell which hangs on the opposite end. R. Descartes - De Homine

6 A global view of signal transduction Lipid-soluble molecules can pass through the cell membrane, but most signals are proteins or peptides and these require more machinery... Soluble factor Ligand Y Receptor Nucleus Cell

7 Receptors can transduce signals in a variety of ways

8 Receptor phosphorylation and G-protein activation: two major transduction pathways

9 The G-protein motif

10 Some of the functions of G proteins

11 One mode involving phosphorylation: the RTK pathway

12 Downstream activation via RTKs

13 A Ras-mediated pathway

14 The signal transduction pathway in E. coli +ATT -ATT +CH 3 R flagellar motor MCPs A W P~ B -CH 3 MCPs W A P ~ Y CW ATP ATP ~ P ADP Z P i B Y P i

15 Metabolic networks Metabolism : The cellular process by which organic molecules are synthesized or degraded, usually via enzyme-catalyzed reactions The interconnected components and reactions form a network, called the metabolic network P P 2 P 1 : 3 A : A B : A E 1 E 2 E 3 D P P P : B : D : C E 4 C E 2 B E 4 E 5 E 1 E 6 C A C E E 3 6 D E 5 Rather than viewing reactions in isolation, as on the left, we should begin thinking about the underlying structure of the network, as well as the individual reactions, as on the right!

16 The glycolytic reactions Glycolysis: The lysis or splitting of glucose Stage 1 Breakdown of complex molecules Stage 2 Glycolysis Stage 3 Production of NADH and ATP

17 Gene control networks, development, and the French flag problem

18 The map from egg to adult in Drosophila melanogaster

19 The general structure of the gene network in Drosophila

20 The details of the segment polarity gene network

21 The general structure of signal transduction cascades 1. General scheme External Signal Outside Signal Detection Inside Signal Transduction Internal Response Signal Propagation

22 Amplification of signals

23 Adaptation to constant signals A single step change in the signal produces a single response and a return to the basal level of activity, while a sequence of steps produces a sequence of responses. Thus the system both adapts AND maintains sensitivity to further changes in the stimulus.

24 The role of adaptation in the visual system

25 What is it we want to understand about these networks? How do we describe their dynamical behavior? Do we use 1. a continuous state space and deterministic descriptions via ODE s, 2. a Boolean representation of ON/OFF states and logical functions that determine the dynamics, or 3. a stochastic description, in which we follow individual molecules. How do we decide which is the appropriate choice? What are the attractors in the dynamics? Are they steady states or fixed points, periodic attractors, or are they more complicated? How do we define and then compute measures of amplification, sensitivity and gain? How sensitive is the behavior to variations in the parameters? In the structure of the network itself? From a dynamical systems point of view this might be viewed as a question of structural stability ; another way of phrasing this is to ask whether certain features of the dynamics are robust.

26 What are the big questions about these networks? Why have the networks evolved to their present form? To what extent do the physics and chemistry influence the structure of the networks? Said otherwise, if one wants to synthesize Z from A, how many distinct (nontrivial) paths are there? The answer bears on the inverse problem of inferring the network from expression patterns. Are there design principles and evolutionary approaches that can help us understand the observed structure? Many signal transduction systems adapt to constant signals but retain the ability to respond to further changes. What structure in a dynamical system guarantees this characteristic? Can we give a useful precise definition of robustness? How do we determine when parametric robustness suffices and when redundancy is necessary? Should we expect robustness at the individual level, or can population-level feedbacks correct individual-level variances in response to signals?

27 Two views on complex networks... When the number of factors coming into play in a phenomenological complex is too large, scientific method in most cases fails. One need only think of the weather, in which case the prediction even for a few days ahead is impossible. Nevertheless, none doubts that we are confronted with a causal connection whose causal components are in the main known to us. Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature. Albert Einstein Though signalling pathways are often drawn as simple linear chains of events, they are rarely that simple. Frequently there is feedback, cross-talk between pathways and enough branching to make one want to change to a less complicated field of biology. In order to fully understand these pathways, we need a convenient and powerful model to complement the experimental research. Though there have been relatively few attempts to model signalling pathways using computers, it seems likely that this will very soon become a major area of study. J. Michael Bishop, Science 267:1617

28 The mathematical description of chemical reaction networks The stochiometry of a chemical reaction denotes the molar ratios in which molecules are converted into products in that reaction. A k B A + 2 B k C MAK: Rate = ka Rate = kab 2 Here, and unless specified otherwise later, we consider only mass action kinetics (MAK), which means that the rates are monomials in the concentrations, raised to the power given by the stoichiometry, of the components that react together. The stochiometric matrix ν = (ν ij ) provides a mathematical encoding of the stoichiometry and topology of a reaction network reactions P 1 A P P 3 2 B D P 4 P 5 C P 6 ν= reactants

29 The mathematical description of chemical reaction networks... Define the concentration and rate vectors c = c 1 c 2.. c n P (c) = P 1 (c) P 2 (c). P n (c) Then the concentration evolves in time according the differential equation dc dt = νp (c) c(0) = c 0 (1) c 3 c 2 c 1

30 Reaction invariants Suppose that there are vectors ξ k R n, k = 1,..., m such that ν T ξ k = 0 for k = 1,..., m. What does this mean for the dynamics? Clearly d dt < ξk, c >=< ξ k, νp (c) >=< ν T ξ k, P (c) >= 0, i.e., the quantity < ξ k, c > is constant in time. We call this a kinematic invariant, and it affects the evolution of the composition as follows. < ξ k, c >= constant defines a linear manifold, and the intersection of this manifold with the positive orthant of composition space defines the reaction simplex. c 3 Reaction simplex c(0) c 2 Remark: Note that ν T has a nontrivial null space if the number of species exceeds the number of reactions. c 1

31 Enzyme-catalyzed reactions Let E denote an enzyme, let S denote the substrate, let when S binds to E, and let P denote the product k 1 k 2 E + S ES E + P k 1 ES denote the complex formed Here c = (E, S, ES, P ) T and ν = The rank of ν is 2 and therefore dim N (ν T ) = 2, so there are two kinematic invariants. We can choose these as ξ 1 = (1, 0, 1, 0) T and ξ 2 = (0, 1, 1, 1) T. Thus the reaction simplex is two-dimensional. How could we have predicted this a priori? Exercise: Determine how the reaction invariants reflect the conservation of atomic species in the reaction H 2 + O H 2 O.

32 A Cartoon Model of Excitation and Adaptation Signal Change Excitation Adaptation Response dy 1 dτ dy 2 dτ = S(τ) (y 1 + y 2 ) τ E = S(τ) y 2 τ A For example, S(t) could be proportional to the fraction of receptors occupied. y 1 = S 0 τ A τ A + τ E (e τ/τ A e τ/τ E ) y 2 = S 0 (1 e τ/τ A )

33 y 2 (a) y 2 (b) y 1 y 1 S(τ) = 0 S(τ) = S 1 S(τ) = 0 S(τ) = S 1 S(τ) = S 2 If τ E << τ A, then for τ >> τ E, y 1 relaxes to u ẏ 2 satisfies y 1 S 0 e τ/τ A S 0 y 2 (τ) = τ A ẏ 2. du dt + 1 u = 1 ds τ A τ A dt

34 Adaptation, Sensitivity and Gain Response dx dt = f(x, S(t)) R = G(x(t)). At steady state suppose x = X(S); then adaptation requires that dr ds = G x i x i S = xg(x(s)), (D x f) 1 (X(S), S)D S f(x(s), S) = 0 Definitions of gain and sensitivity g 0 dr ds = G x i x i S g LS dr d ln S = Sg 0 g LR d ln R ds = 1 R g 0 g LL d ln R d ln S = S R g 0 Thus the steady-state gain vanishes in a system that adapts perfectly!

35 Sensitivity of transient solutions x S = Φ(t, 0) x S (0) + t 0 Φ(t, τ)d S f(x(τ), S(τ))dτ ḡ = max [0, ) g(t) Reported bacterial gains range from g = 6 to g = 55

36 Example Cartoon model Step to θ 1 at t = 0 dy 1 dt dy 2 dt = θ(t) (y 1 + y 2 ) τ E = θ(t) y 2 τ A y 1 = θ 1τ A τ A + τ E (e t/τ A e t/τ e ) Thus y 1 θ 1 = τ A τ A + τ E (e t/τ A e t/τ E ) g 0 = y 1 θ 1 g LS = θ 1 g 0 = y 1 g LL = θ 1 y 1 g 0 1 g LR = g 0 R g 0 1 θ 1

37 What do we mean by gene expression

38 The Jacob-Monod model

39 The four basic modes of transcription control

40 The lac operon

41 The effect of lactose

42 Eucaryotic transcription control

43 The loci of control and modification

44 Four types of models Boolean models Mixed models... in which everything is either ON or OFF and the state space is a finite set... in which some things are either ON or OFF, while others vary continuously Deterministic continuous models Stochastic models... in which everything varies continuously, the state space is a subset of R n, and the dynamics are deterministic... in which we recognize that reactions occur one molecule at a time

45 The Boolean approach

46 One realization of the not if or C and (not D)

47 A general description of a gene control pathway based on feedback

48 The governing equations for a continuous description ds 1 = R(S n+1 ) A N T 1 (S 1, S 2 ) dt V k 1 S 1 N ds 2 = A N T 1 (S 1, S 2 ) k 2 S 2 dt V C ds 3 = dt k 2 S 2 k 3 S 3 ds 4 = dt k 3 S 3 k 4 S 4. ds j = dt k j 1 S j 1 k j S j, j = 5,..., n 1. ds n = dt k n 1 S n 1 k n S n A N T n (S n, S n+1 ) V C ds n+1 dt where k j = k j + k j for 4 j n 1. = A N V N T n (S n, S n+1 ) k n+1 S n+1

49 The control functions for inducible and repressible systems R + ps RS p, R + O OR, R = repressor, O = operator, and S = effector. Fraction of operator regions free of repressor: K = 1 + K 2 R t > 1 K 1 = RS p /R S p K 2 = OR/R O R t = R + RS p = R(1 + K 1 S p ) O t = O + OR = O(1 + K 2 R) f(s) = O O t = 1 + K ts p K + K 1 S p (2)

50 For a repressible system R + ps K 1 RS p RS p + O K 2 ORS p In this case f(s) = 1 + K 1S p 1 + K 1 KS p (3)

51 Quantitative characterization of some gene control systems Enzyme Effector p K 1 K 2 R t Inducible β-galactosidase Isopropylthio M galactoside Histidine-NH 3 -lyase Imadizole M 2 26 propionate Urocanase Histidine M Mannitiol Ribitol 3.13 Repressible dehydrogenase IMP dehydrogenase Guanine 0.91 XMP aminase Guanine 0.68 Alkaline a From Yagil and Yagil (1971). P O M