Modelling in Biology
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1 Modelling in Biology Dr Guy-Bart Stan Department of Bioengineering 17th October 2017 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
2 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Essential features of a modelling approach Isolate your system of interest. What is important? This defines your system of interest What can be measured? What are the observables? This defines the outputs of the system. What can be controlled or acted upon? This defines the inputs of the system. Inputs System of interest Outputs Outside Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
3 Modelling of the system of interest Typically, the model is composed of variables independent, e.g., time t 1 indep. var.: ODEs, e.g., time t more than 1 indep. var.: PDEs, e.g., time t and space (, y, z) dependent e.g., concentrations functions of time {[E](t), [S](t), [P](t)} parameters not dependent on independent variables can be varied/changed under eperimental conditions constants fied, e.g., Avogadro constant, gravitational constant Different types of models Dr Based Guy-Barton Stan these (Dept. of concepts, Bioeng.) different Modelling types in Biology of models can be17th built. October / 77 Types of models Continuous the independent variables are continuous ODEs, PDEs Deterministic var., param. and const. do not contain randomness they are defined by a unique function Linear ẋ = d = k Linear ODE Autonomous Without control input: ẋ = k Constructive mechanistic or deductive also called equation-based or (first) principle-based Discrete the independent variables are discrete Difference equations Stochastic dynamics contain an element of randomness e.g., SDEs Nonlinear ẋ = d = k + 3 Nonlinear ODE Non-autonomous With control input: ẋ = k + u Data-driven phenomenological or inductive Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
4 Summary Hybrid models Continuous (indep. var. continuous) Discrete (indep. var. discrete) (time) ODE SDE (randomness) (time + space) PDE Fokker-Planck (Kolmogorov forward) equation Discretisation Difference eqn. Finite Element Methods Continuous Discrete ODEs PDEs Deterministic (L or NL) SDEs Stochastic (L or NL) Difference equations Deterministic (L or NL) Stochastic (L or NL) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
5 A deterministic, continuous, linear model of order 1 Consider ẋ(t) = d(t) = k(t) Linear ODE. For k > 0, this is known as the Malthusian population growth with k denoting the growth rate per cell. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Analytical solution of 1 st order linear ODEs Consider the model: ẋ = d = k, (0) = 0 (1) Its solution is given by (t) = 0 e kt where 0 = (0) (the initial condition). k>0 0 k =0 k<0 t Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
6 Numerical solution of ODEs: the Euler algorithm d = k (t + t) (t) lim t 0 t = k(t) Suppose t is fied to a particular value h (doing this is called discretising the continuous ODE model and h is called the discretisation step). We then have: (t + h) (t) k(t) h (t + h) (t) + hk(t) (2) Eq. (2) is know as the Euler algorithm. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Linear difference equation (t + h) = (t) + hk(t) is a discrete-time model which can also be looked at as a linear difference equation by taking h = 1, and defining for ease of notation t = (t): (or equivalently t+1 t = (α 1) t.) Its non-zero solution is given by where 0 is the initial condition. t+1 = (1 + k) }{{} t = α t (3) α t = 0 α t Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
7 Phase plane The phase plane is a representation that eliminates time as an eplicit variable. It is very useful for obtaining a qualitative understanding of the long-term or asymptotic behaviour of nonlinear ODE models (for which analytical solutions cannot typically be found). Consider ẋ = k ẋ = k k>0 ẋ = k k<0 0 0 =1 0 = 1 0 ẋ = k Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 k =0 Bifurcation diagram 0 We can summarise the information obtained through the phase plane stability analysis on a bifurcation diagram, i.e., a diagram giving the long-term (i.e., asymptotic) behaviour of the system when a parameter is varied. Here the parameter for the ODE model ẋ = k is k. ẋ = k k>0 + k<0 0 k Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
8 SDEs of order 1 Consider a stochastic version of the Malthusian growth model: d = k + η (4) where η is a random variable that represents some uncertainties or stochastic effects perturbing the system. Eq. (4) is known as a Langevin equation. Eq. (4) can also be rewritten as d = [k] + η }{{} σdw where w represents a standard (one-dimensional) Wiener process (also called Brownian motion) SDEs such as (5) are typically solved numerically through discretisation using the Euler algorithm: (t + t) = [1 + k t](t) }{{} deterministic part ( + σ ) t randn } {{ } stochastic part Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 (5) SDE of order 1 (cont ) σ 1 k 0 run 1 run 2 t σ 2 > σ 1 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 σ
9 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Nonlinear ODE models of order 1 First order nonlinear ODE models are written under the generic form: ẋ = f (), R, f ( ) : R R, smooth function (6) Finding the analytical solution of (6), i.e., finding (t, 0 ), is, in general, no longer possible unless a closed form solution can be obtained for 1 f () d =. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
10 Non-Malthusian population growth: the logistic equation We consider the non-malthusian population growth model in which the reproduction rate takes into account the competition for resources. Consider that (t) represents the number of cells at time instant t. =f () { ( }}{ ẋ = r 1 ) = }{{ k } Resources ( r 1 ) }{{ k } non-constant growth rate per cell = r }{{} growth rate r 2 k }{{} death rate (7) In this particular case and rather eceptionally, a closed form solution to (7) can be found: (t) = k C e rt, C = 0 k 0 This solution indicates that k as t. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The logistic equation Time solution: (t) = k C e rt, C = 0 k 0 k Carrying capacity k 2 0 Infleion point t Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
11 Stability analysis of the logistic equation ẋ = r ( 1 ) k Fied points and flow: Fied points: ẋ = = 0 f ( ) = 0 { ( ) Here, f ( ) = r = 0 1 = 0 k = k Flow: { 0 < < k ẋ > 0 > k ẋ < 0 Phase plane: ẋ vs ẋ = r k 2 + r Infleion point for (t) k 2 k Asymptotic stability of fied points: = 0 is unstable = k is asymptotically stable, i.e. stable attractive Attractors: = k Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Stability analysis of nonlinear ODE models of order 1 Consider a nonlinear ODE model of order 1: ẋ = f (), R, f ( ) : R R, smooth function 1 Global stability analysis (only for models of order 1) Find all the fied points: { : f ( ) = 0} and put them on the phase line of the plot ẋ vs. Find the flow between the fied points and indicate them on the phase line of the plot ẋ vs. Conclude what the stability of the fied point(s) is. Find the long-term behaviour of the system, i.e., its attractors. 2 Local/linear stability analysis (possible for all orders) Find the fied points. Linearise the dynamics around each fied point. Study the stability of the corresponding linear systems (eig(a)). Link together the local stability information around each fied point to establish a complete picture of the attractors. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
12 Linear stability analysis of ODE models of order 1 Find the fied points of the system: f ( ) = 0. Eamine the close neighbourhood of the fied points, i.e., analyse the local stability of the fied points by considering small perturbations around them. ẋ = r k 2 + r Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Linear stability analysis of ODE models of order 1 Consider the dynamics of the system when is close to the fied point, i.e., consider ẋ = f () when = + ξ with ξ = ( ) small, i.e., ξ 1 : d = dξ = f ( + ξ) = f ( ) }{{} =0 + df d = ξ + O ( }{{} ξ 2) }{{} small H.O.T. ( very small ) (Taylor series epansion) So, we have: dξ df d = ξ (linear system) ξ(t) ξ 0 e df d = t Local stability analysis (only two possibilities): df d = > 0 ξξ > 0 ξ = is unstable df d = < 0 ξξ < 0 ξ = is locally asymptotically stable, i.e., locally stable and attractive Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
13 Linear stability analysis of the logistic equation For ẋ = r ( 1 ) df k, we have d = r 2r k df d =0 = r > 0 ξξ > 0 ξ = 0 is unstable df d =k = r < 0 ξξ < 0 ξ = k is locally asymptotically stable, i.e., locally stable and attractive Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Bifurcations for nonlinear ODE models of order 1 Consider: ẋ = f (, r) where r is a parameter and f ( ) : R R R is a smooth function. Bifurcation A bifurcation occurs when a change in the parameter(s) of the model produces qualitative (or large ) changes in the long-term behaviour (of the attractors) of the system, e.g., : the number of attractors (e.g., fied points) changes, the type of attractors changes (e.g., from fied point to limit cycle), the stability of attractors (e.g., fied points or limit cycles) changes. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
14 Saddle-node Bifurcation ẋ = r + 2, R Consider different values for the parameter r: ẋ r<0 ẋ ẋ r =0 r>0 r r Saddle-node bifurcation diagram: = p r unstable r stable = p r Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Transcritical Bifurcation ẋ = r 2 = (r ), R Consider different values for the parameter r: ẋ ẋ ẋ r<0 r =0 r>0 r r Transcritical bifurcation diagram: = r r =0 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
15 Pitchfork Bifurcation ẋ = r 3 = ( r 2), R Consider different values for the parameter r: ẋ r<0 ẋ r =0 r>0 ẋ r r Pitchfork bifurcation diagram (supercritical): = p r r =0 = p r Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Summary of behaviours for NL ODE models of order 1 Motions (solutions) are on the real line, i.e., R Attractors are either the fied points or ± (no oscillatory or other types of behaviour) Three types of bifurcation can occur: Saddle node Transcritical Pitchfork (subcritical or supercritical) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
16 Enzymatic reactions and the law of mass action E ES E S P Enzymatic reaction: k 1 E + S FGGGGG GGGGGB ESGGGAE + P k 1 Law of mass action: For a simple enzymatic reaction we have 4 species 4 ODEs k 2 d[es] d[e] d[s] d[p] = k 1 [E][S] k 1 [ES] k 2 [ES] (8) = k 1 [E][S] + k 1 [ES] + k 2 [ES] (9) = k 1 [E][S] + k 1 [ES] (10) = k 2 [ES] (11) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Elimination of variables model reduction through time scale separation 1 Conservation laws (8) + (9) d[es] + d[e] = 0 [ES] + [E] = [E] 0 (12) (9) - (10) - (11) d[e] d[s] d[p] = 0 [E] = [S] + [P] + κ (13) 2 Quasi-stationary approimation (time scale separation) d[s] with d[es] 0 d[p] [S] V ma K M + [S] V ma = k 2 [E] 0, K M = k 1 + k 2 k 1 (the Michaelis-Menten equation) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
17 The Michaelis-Menten equation d[s] d[p] [S] V ma K M + [S] (the Michaelis-Menten equation) [Ṡ] V ma 2 V ma K M [S] Concentration S ES T ime P E Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Enzymatic cooperative reactions The Hill equation E ES E S S S S S P A model for the enzymatic reaction with cooperativity is: k 1 E + ns FGGGGG GGGGGB ESGGGAE + P k 1 where ES represents the enzyme-n-substrates comple and n is called the cooperativity coefficient. Law of mass action: 4 species 4 ODEs k 2 d[es] d[e] d[s] d[p] = k 1 [E][S] n k 1 [ES] k 2 [ES] (14) = k 1 [E][S] n + k 1 [ES] + k 2 [ES] (15) = n ( k 1 [E][S] n + k 1 [ES]) (16) = k 2 [ES] (17) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
18 The Hill equation Using a similar model reduction approach as for the non-cooperative enzymatic reactions we saw before (Michaelis-Menten), it is easy to see that the following 1 st order nonlinear ODE model is obtained: with d[s] d[p] [S] n V ma K M + [S] n V ma = nk 2 [E] 0, K M = k 1 + k 2 k 1 [Ṡ] n KM (the Hill equation) V ma 2 [S] V ma Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The Hill equation: effect of the cooperativity coefficient n The Hill function is defined as h() = V ma K M +. The effect of the Hill n coefficient n is illustrated hereafter for V ma = 1 and K M = 1: n 1 n = n =1 0.5 Increasing n This is very useful for a cell which can then use this type of step-regulated reaction as a switch since for low concentrations (i.e., n K M ) nothing happens, while for high concentrations (i.e., > n K M ) the enzymatic reaction happens at its maimal rate V ma. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
19 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 A chemical eample of a linear ODE model of order 2 Consider the chemical reaction: X FGGGG GGGGB k Y k Using the law of mass action, the corresponding ODEs write: [Ẋ ] = k[x ] + k[y ] (18) [ Y ] = k[x ] k[y ], k > 0 (19) ( ) ( ) 1 [X ] To solve (18)-(19) analytically, we define the vector = = 2 [Y ] and rewrite the equation under the form ẋ = A. We then use a change of variables in order to diagonalise the matri A. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
20 Diagonalisation, eigenvalues and eigenvectors The system of equations (18)-(19) can be rewritten as: ) ( ) (ẋ1 1 1 = k ẋ }{{} =A ( 1 2 ) ẋ = ka (20) To solve (20), we diagonalise A, i.e., we find its eigenvalues and eigenvectors. 1 Eigenvalues: Solutions of det(a λi ) = 0. Here, we have: λ 1 = 0 and λ 2 = 2. 2 Eigenvectors (normalised): Solutions ( of ) Av = λv, for each eigenvalue λ. Here, we have: v 1 = corresponding to λ 1 1 = 0 ( ) and v 2 = corresponding to λ 1 2 = 2. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Diagonalisation, eigenvalues and eigenvectors (cont ) From the eigenvectors of A, we construct a new matri V having the eigenvectors of A as columns: V = v 1 v 2 = 1 2 ( We then have (theorem on diagonalisation of matrices): ( ) ( ) V 1 λ AV = Λ = = 0 λ ) (21) Now, recall the initial model was ẋ = ka. Multiplying this latter equation by V 1 on the left gives: V 1 d = d (V 1 ) }{{} =X dx = k } V 1 {{ AV} (V 1 ) }{{} =Λ =X = kλx = kλx, X = V 1 (22) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
21 Diagonalisation, eigenvalues and eigenvectors (cont ) ) (Ẋ1 Ẋ2 ( ) ( ) 0 0 X1 = k 0 2 X 2 ( ) ( ) λ1 0 X1 = k 0 λ 2 X 2 (23) {Ẋ1 = kλ 1 X 1 Ẋ 2 = kλ 2 X 2 { X 1 (t) = X 1 (0)e kλ 1t X 2 (t) = X 2 (0)e kλ 2t The last step is to transform back into the original coordinates using X = V 1 which implies = V X. Using = V X, i.e., ( ) = v 1 v 2 X1 = v X 1 X 1 + v 2 X 2, we obtain 2 (t) = v 1 X 1 (t) + v 2 X 2 (t) (t) = v 1 X 1 (0)e kλ 1t + v 2 X 2 (0)e kλ 2t Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The mass-spring-damper system Let us consider the mass-spring-damper system: η κ m OR κ η m for which the equation of motion is mẍ + ηẋ + κ = 0 (24) To solve (24), we put the model in the form ẋ = A and diagonalise A: { (ẋ ) ) ẋ = y ẍ = ẏ = κ m η m y ẏ ( 0 1 = κ m η m }{{} =A ( ) y The eigenvalues of A are λ ± = η m ± η 2 m 2 4 κ m 2. The general solution is thus (t) = c + e λ +t + c e λ t where c ± are proportional to the eigenvectors associated with λ ±. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
22 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Stability analysis of nonlinear ODE models of order 2 Consider a nonlinear ODE model of order 2: ẋ = f (), ) (ẋ1 R 2 = ẋ 2 ( ) f1 ( 1, 2 ), f ( ) : R 2 R 2, smooth funct f 2 ( 1, 2 ) 1 Global stability analysis (difficult for models of order 2) 2 Local stability analysis (possible for all orders) Find the fied points: { : f ( ) = 0}. Linearise the dynamics around each fied point. Study the stability of the corresponding linear systems. Draw the local flows around each fied point: Try to link together the local stability information around each fied point to establish a global picture of the attractors in the state space. Nullclines: the curves in the phase planes corresponding to individual first derivatives being zero (ẋ 1 = 0 or ẋ 2 = 0), i.e., the curves f 1 ( 1, 2 ) = 0 and f 2 ( 1, 2 ) = 0. Trajectories in the phase plane cannot cross, ecept at the fied points. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
23 Linearisation of ODE models of order 2 Find the fied points, { i.e., the points s.t. ẋ = f ( f 1 (1 ) = 0, 2 ) = 0 f 2 (1, 2 ) = 0 Linearise the dynamics around each fied point (using Taylor): Consider ẋ = f () with = + ξ where ξ R 2 s.t. ξ 1: ξ = ẋ = f ( + ξ) = f ( ) + }{{} =0 We thus obtain: f 1 ( 1, 2 ) 1 f 2 ( 1, 2 ) 1 = = f 1 ( 1, 2 ) 2 f 2 ( 1, 2 ) 2 = = } {{ } J( ) ξ + O ( ξ 2) }{{} H.O.T. ξ = J ( ) ξ (25) where J ( ) is the Jacobian matri evaluated at the fied point, i.e., a constant matri whose (i, j) element is J ij ( ) = f i () j =. Study the stability of (25) at each fied point Diagonalise J ( ) (i.e., ξ = V 1 ξ) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Diagonalisation of the Jacobian for models of order 2 Consider ( ) J ( a b ) = c d The eigenvalues of J ( ) are given by solving which gives the algebraic equation where det (J ( ) λi ) = 0 λ 2 (a + d) λ + (ad bc) = 0 }{{}}{{} =τ = τ is the trace of J ( ), i.e., the sum of the diagonal elements of J ( ) (= λ + + λ ) is the determinant of J ( ) (= λ + λ ) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
24 Diagonalisation of the Jacobian for models of order 2 (cont ) Eigenvalues: λ 2 τλ + = 0 λ ± = τ ± τ Therefore, diagonalising J ( ), we can see that the general solution is of the form: ξ(t) = c + e λ +t + c e λ t, ξ(t), c +, c R 2 (26) where λ ± are the eigenvalues of J ( ) and c ± are proportional to the corresponding eigenvectors. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Local stability analysis for models of order 2 The general solution for a linear ODE of order 2 is: ξ(t) = c + e λ +t + c e λ t, λ ± = τ ± τ 2 4, 2 c ± eigenvec. assoc. with λ (27) The local behaviours are dictated ( by the signs of τ(= λ + + λ ), (= λ + λ ), and τ 2 4 = (λ + λ ) 2). 1 > 0: τ 2 4 < τ τ > 0: 1 τ 2 4 > 0: λ + > λ > 0 (A) 2 τ 2 4 < 0: λ ± comple conjugate with pos. real part (B) τ < 0: 1 τ 2 4 > 0: λ < λ + < 0 (C) 2 τ 2 4 < 0: λ ± comple conjugate with neg. real part (D) τ = 0: λ± purely imaginary (E) 2 < 0: λ < 0 < λ + (F) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
25 Local stability analysis for models of order 2 (cont ) In the diagonalised coordinates, i.e., for ξ = V 1 ξ: ξ 1 (t) = ξ 1 (0)e λ +t, ξ 2 = ξ 2 (0)e λ t, λ ± = τ ± τ (A) λ + > λ > 0: Eponential growth in both directions: Repelling or unstable node (B) λ ± comple conjugate with pos. real part: ξ 1,2 (t) = ξ 1,2 (0)e τ 2 t e ±iωt τ : Unstable spiral with τ > 0, ω = (C) λ < λ + < 0: Eponential decay in both directions: Attracting or stable node (D) λ ± comple conjugate with neg. real part: ξ 1,2 (t) = ξ 1,2 (0)e τ 2 t e ±iωt τ : Stable spiral with τ < 0, ω = (E) λ ± purely imaginary: ξ 1,2 (t) = ξ 1,2 (0)e ±iωt with ω = : periodic oscillations: Center (F) λ < 0 < λ + : ep. decay in one dir. and ep. growth in the other: Saddle point Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Summary of the possible local behaviours for models of order 2 τ τ 2 4 > 0 (A) Unstable nodes τ 2 4 =0 (F) Saddle points Unstable (B) spirals (E) Centers (D) Stable spirals (C) Stable nodes τ 2 4 =0 τ 2 4 > 0 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
26 Linear (or harmonic) oscillations (centers) { ẋ = y ẏ = Linear (or harmonic ) oscillations have 2 serious limitations: 1 They are fragile or non-robust to small perturbations in the model 2 The oscillation characteristics (amplitude and phase) depend on the initial condition Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Limit cycles A stable limit cycle is a periodic trajectory which attracts other solutions to it (at least those starting close to the limit cycle). 2 1 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
27 Limit cycles: an eample ẋ = ωy + ( µ 2 y 2) (28) ẏ = ω + y ( µ 2 y 2) (29) where µ is a parameter while ω 0 is a constant. Fied points: { ωy + ( µ 2 y 2) = 0 ω y ω + y ( µ 2 y 2) = 0 = ω y y = 0 (, y ) = (0, 0) Linearisation around (0, 0): (( µ J(, y) = 2 y 2) ) + ( 2) ( ω 2y ω 2y µ 2 y 2) + y( 2y) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Limit cycles: an eample (cont ) Local stability analysis of (0, 0): J(0, 0) = ( ) µ ω ω µ λ ± = µ ± iω τ = 2µ = µ 2 + ω 2 > 0 τ 2 4 = 4µ 2 4µ 2 4ω 2 < 0 τ Unstable spirals τ 2 (B) 4 < 0 (E) Centers (D) Stable spirals τ 2 4 =0 µ>0, Unstable spiral µ =0, Center µ<0, Stable spiral Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
28 Global stability analysis for the limit cycle eample (rare) To perform the global analysis eplicitly (which typically is very hard to do; the eample we have chosen here is an eception in that respect), we rewrite (28)-(29) in polar coordinates: y 0 r θ r y θ r 2 = 2 + y 2 tan(θ) = y i.e., transform ẋ = f () with = This gives: ( ) y into ṗ = F (p) with p = ( ) r. θ ṙ = r ( µ r 2) (30) θ = ω (31) Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Global stability analysis for the limit cycle eample (cont ) ṙ µ<0 ṙ µ =0 ṙ µ>0 µ µ r r r y 0 r θ Hopf bifurcation at µ =0 y 0 µ Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
29 Eample of a limit cycle in a model of order 3 State space of a SINGLE oscillator for k p = e 01 State space of a SINGLE oscillator for k p = e X X X X Time evolution of the state variables of the cart pole system dot dot Time evolution of the state variables of the cart pole system Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 dot dot 6 6 state variable The Poincaré-Bendison Theorem 0 0 state variable D time y time A loose statement of the Poincaré-Bendison Theorem Suppose ẋ = f (), R 2, is a continuously differentiable vector field and there eists a bounded subset D of the phase plane such that no trajectory can eit D, and there are no fied points inside D. Then, there eists at least one limit cycle in D and any trajectory that enters D converges to a limit cycle. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
30 The Poincaré-Bendison Theorem: How can we find D? We need to show that: T g(, y)f (, y) < 0, Ā and T b(, y)f (, y) > 0, B y D = A B f() f() g ẋ = f(), = y g D = A B = {(, y) R 2 : g(, y) 0,b(, y) 0} Ā = {(, y) R 2 : g(, y) 0} B = {(, y) R 2 : b(, y) 0} Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The Poincaré-Bendison Theorem: an eample Consider the second order model in eample in (28)-(29) for µ > 0: We thus have: f (, y) = ẋ = ωy + ( µ 2 y 2) ẏ = ω + y ( µ 2 y 2) ( ( ωy + µ 2 y 2) ) ω + y ( µ 2 y 2). Consider the function G(, y) = 2 + y 2 R 2. Therefore, G = Now, define D = { (, y) : g(, y) = 2 + y 2 R 2 = 0 }. For any point (, y) D, we thus have: T G(, y)f (, y) = 2R 2 ( µ R 2) { < 0 if R > µ > 0 if R < µ, y ( ) 2. 2y D Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
31 Summary of behaviours for ODE models of order 1 and 2 Models of order 1 Attractors: fied points, or Local (linear) and global stability analysis are equivalent Bifurcations: Saddle node, Transcritical, or Pitchfork Models of order 2 Attractors: fied points, limit cycles, or Local stability analysis (linearisation) around fied points global stability analysis Bifurcation: all those of order 1 + Hopf + others Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
32 Behaviours for ODE models of order 3 and higher ẋ = f (), R d, d 3, f ( ) : R d R d, smooth function (32) Behaviours for ODE models of order 3 and higher In one sentence: everything that happens in lower order models + 2 other phenomena: quasi-periodicity deterministic chaos Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Summary for systems of order 1, 2, and 3 and higher 1D 2D 3D and higher Attractors Only F.P. or ± Same as 1D + limit cycles Same as 2D + quasi-periodic attractor + chaotic attractor Behaviours Decay or eplosion Same as 1D + robust periodic oscillations Same as 2D + quasiperiodicity + chaos Bifurcations Saddle-Node Transcritical Pitchfork Same as 1D + Hopf + others Same as 2D + many more Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
33 1 Introduction 2 Linear models of order 1 3 Nonlinear ODE models of order 1 4 Linear ODE models of order 2 and higher 5 Nonlinear ODE models of order 2 6 Nonlinear ODE models of order 3 and higher 7 Modelling gene regulation networks Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The central dogma Transcription and mrna polymerase Translation and Ribosome Translation and Proteins Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
34 Constitutive gene epression As we have seen, the central dogma can be summarized as: Gene GGGGGGA mrna GGGGGA Protein Transcr. Transl. When gene epression is unregulated, it is said to be constitutive, and the gene is always on. Using the law of mass action 1, a model for constitutive epression is given as: where m = [mrna] and p = [Protein] ṁ = k 1 d 1 m (33) ṗ = k 2 m d 2 p (34) k 1 is the constitutive transcription rate. d 1 is the mrna degradation rate. k 2 is the translation rate. d 2 is the protein degradation rate. 1 This is based on empirical studies since strictly speaking it does not really make sense to use the law of mass action for gene epression. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Gene transcription regulation At the transcription level, gene epression can be controlled by certain proteins called transcription factors: Transcription Factor (TF) (activator or repressor) TF binding site Promoter RBS Protein coding region Terminator Gene DNA Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
35 Gene transcription regulation by activators Consider the case of a gene whose transcription is activated by the cooperative binding of activators to the transcription factor binding site of the gene. Activators Gene mrna Protein The following model is commonly used to describe activator controlled gene transcription: A n ṁ = k 1 K n + A n d 1m (35) ṗ = k 2 m d 2 p (36) where m = [mrna], p = [Protein], A = [Activator], k 1 = maimal transcription rate, K = activation coefficient, n = Hill coefficient (= number of activators that need to cooperatively bind the promoter to trigger the activation of gene epression). Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Gene transcription regulation by repressors Consider the case of a gene whose transcription is repressed by the cooperative binding of repressors to the transcription factor binding site of the gene. Repressors Gene mrna Protein The following nonlinear ODE model describes repressor controlled gene transcription: K n ṁ = k 1 K n + R n d 1m (37) ṗ = k 2 m d 2 p (38) where m = [mrna], p = [Protein], R = [Repressor], k 1 = maimal transcription rate, K = repression coefficient, n = Hill coefficient (= number of repressors that need to cooperatively bind the promoter to trigger the inhibition of gene epression). Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
36 Auto-activation and auto-inhibition + or - Gene mrna Transcription Factor (protein) ; ; ṁ = k 1 f (p) d 1 m (39) ṗ = k 2 m d 2 p (40) where f (p) = f + (p) = pn K n +p (monnotonically increasing Hill function) for n an auto-activating action of the transcription factor p, and f (p) = f (p) = 1 f + (p) = K n K n +p (monotonically decreasing Hill n function) for an auto-inhibiting action of the transcription factor p. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 Auto-activation p n ṁ = k 1 K n + p n d 1m (41) ṗ = k 2 m d 2 p (42) m ṗ =0 m ṗ =0 ṁ =0 ṁ =0 n =1 n 2 p p Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
37 Auto-repression K n ṁ = k 1 K n + p n d 1m (43) ṗ = k 2 m d 2 p (44) m ṗ =0 m ṗ =0 n =1 n 2 ṁ =0 ṁ =0 p p Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 The toggle switch LacI TetR ṁ L = k L,1 K n T T K n T T + pn T T ṗ L = k L,2 m L d L,2 p L ṁ T = k T,1 K n L L K n L L + pn L L ṗ T = k T,2 m T d T,2 p T d L,1 m L d T,1 m T where m L (resp. m T ) is the concentration of LacI (resp. TetR) mrna, and p L (resp. p T ) is the concentration of LacI (resp. TetR) protein. Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
38 The toggle switch (cont ) Using a quasi-stationary assumption for the mrna dynamics, i.e., ṁ L 0 and ṁ T 0, we obtain a model of order 2 whose equations are: ṗ L = k L,2 k L,1 d L,1 ṗ T = k T,2 k T,1 d T,1 K n T T K n L L K nt T + pn T T K nl L + pn L L d L,2 p L (45) d T,2 p T (46) For eample, for n L = n T = 2, k L,1 = k T,1 = 10, and all other parameters equal to 1 the phase plane looks like this: p T ṗ L =0 ṗ T =0 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77 p L The repressilator: a synthetic genetic oscillator LacI TetR λ ci The corresponding model is of order 6 and can be written as (after non-dimensionalisation): ṁ i = m i + α 1 + pj n + α 0 (47) ṗ i = β (p i m i ) (48) where (i, j) = {(LacI, ci ), (TetR, LacI ), (ci, TetR)}. For certain values of the parameters, this system ehibits limit cycle oscillations Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
39 Time evolution of protein concentrations for the repressilator Time evolution of protein concentrations Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October / 77
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