System Biology - Deterministic & Stochastic Dynamical Systems

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1 System Biology - Deterministic & Stochastic Dynamical Systems System Biology - Deterministic & Stochastic Dynamical Systems 1

2 The Cell System Biology - Deterministic & Stochastic Dynamical Systems 2

3 The Cell System Biology - Deterministic & Stochastic Dynamical Systems 3

4 Transcription System Biology - Deterministic & Stochastic Dynamical Systems 4

5 Evolution Design System Biology - Deterministic & Stochastic Dynamical Systems 5

6 Transcription System Biology - Deterministic & Stochastic Dynamical Systems 6

7 Hill Formalism - Single TF Standard equation for change of concentration of a protein: d[t ] dt = P T [T ] τ T [T ]-transcription factor (TF) concentration, P T -production function, τ T - T half life T binds to operator site O with rate k + and unbinds with rate k. [T ] + [O] k k + [TO] The corresponding rate equation is d[t ] = k + [T ][O] + k [TO] dt At equilibrium k + [T ][O] + k [TO] = 0 thus k = [T ][O] [TO] System Biology - Deterministic & Stochastic Dynamical Systems 7

8 [TO] = [T ][O] [k] Total concentration of operator is [O total ] = [O] + [TO]; the bound fraction is: [TO] [O total ] = [TO] [TO] + [O] = [T ][O] k [T ][O] + [O] k = [T ] k 1 + [T ] k If T is an activator the production is proportional to the bound fraction ( [T ] ) h d[t ] = α k dt ( [T ] ) [T ] h τ T 1 + k If T is a repressor the production is proportional to the unbound fraction d[t ] dt 1 = α ( [T ] ) [T ] h τ T 1 + k System Biology - Deterministic & Stochastic Dynamical Systems 8

9 Hill Formalism - Multiple TF Two transcription factors T i, T j activate T following an AND logic d[t ] dt ( [Ti ]) h ( [Tj ]) h = k ( [Ti ]) k h ( [Tj ]) [T ] h τ T k k Two transcription factors T i, T j activate T following an OR logic d[t ] dt ( [Ti ]) h ( [Tj ]) h = k ( [Ti ]) + k h ( [Tj ]) [T ] h τ T k k System Biology - Deterministic & Stochastic Dynamical Systems 9

10 Shea-Akers Formalism - Multiple TF Transcription can be modeled from a statistical point of view. Depending on presence or absence of TF and/or RNAp the operator can be in various states denoted s. Each state has a statistical weight: Z(s) =number of ways the state can be realized e EnergyOfState k b T For a state where i TF are bound the statistical weight for any of the possible s states is given by Z(s) = e G(s) k b T i [T i] [T i ]-concentration of bound TF, e G(s) k b T - parameter for binding affinity, related to loss of free energy at binding. Z(s)is normalized such that the weight of the state with nothing bound is 1 i.e. Z 0 = 1 System Biology - Deterministic & Stochastic Dynamical Systems 10

11 Shea-Akers Formalism - Multiple TF The concentration-dependent factor [T i ] reflects the entropy loss during transitions from freely moving factors to bound factors The total statistical weight (partition sum) is: Z = s Z(s) = Z(on) + Z(off ) The probability of a state on is given by appropriately normalized ratio P(on) = Z(on) Z Where Z(on) = e G(s) k b T Z(off ) = 1 i [T i] and e G(s) k b T = k if s depend only on T i P(on) = P(on) = G(s) e k b T i [T i ] 1+e G(s) k b T i [T i ] i [T i ] k 1+ i [T i ] k System Biology - Deterministic & Stochastic Dynamical Systems 11

12 The bound fraction is: P T = α Z(bound) Z There are 5 possible states of the system thus partition sum is: Z = 1 + p k p + A k A + A p k Ap + R k R all k are dissociation constants. P T = α p k p + A p k Ap 1 + p k p + A k A + A p k Ap + R k R System Biology - Deterministic & Stochastic Dynamical Systems 12

13 LIF BMP4 Deterministic approach. Shea-Ackers equation d[n] dt d[os] dt d[fgf ] dt d[g] dt = k 0 [OS](c 0 + c 1 [N] 2 + k 0 [OS] + c 2 LIF ) (1 + (k 0 [OS](c 1 [N] 2 + k 0 [OS] + c 2 LIF + c 3 [FGF ] 2 )) + c 4 [OS][G] 2 ) γ[n], = α + = = (e 0 + e 1 [OS]) (1 + e 1 [OS] + e 2 [G] 2 γ[os], (1) ) (a 0 + a 1 [OS]) γ[fgf ], (1 + a 1 [OS] + a 2 I 3 ) (b 0 + b 1 [G] 2 + b 3 [OS]) (1 + b 1 [G] 2 + b 2 [N] 2 + b 3 [OS]) γ[g], System Biology - Deterministic & Stochastic Dynamical Systems 13

14 Complex Formation A transcription factor can form complexes with other proteins. Sometimes only the complex participates in the transcription. A free andr free are transcription factors that form complex [A R] At equilibrium d[r free ] dt [A free ] + [R free ] k k + [A R] = 0 [R free ][A free ] k d [A R] = 0 (2) Where k d is the affinity constant. The smaller k d the more complex k d = [R free][a free ] [A R] The total amount of proteins are: [A total ] = [A free ] + [A R] (3) [R total ] = [A free ] + [A R] (4) System Biology - Deterministic & Stochastic Dynamical Systems 14

15 Complex Formation From equations 2, 3 and 4 one obtains the quadratic equation for the complex. [A R] 2 ([R total ] + [A total ] + k d )[R A] + [R total ][S total ] = 0 The solution is: [A R] = [R total] + [A total ] + k d 2 ( [R total] + [A total ] + k d ) 2 2 [R total ][A total ] Why did we chose the solutions with minus sign? if k d is large one gets no complex if k d is small the complex is min([r total ], [A total ]) System Biology - Deterministic & Stochastic Dynamical Systems 15

16 Deterministic Model Limitations dx i dt = f i (X 1...X N ) X-number of molecules, f-interaction function, N-types of molecules assumes that chemical reacting systems are continuous (they re not) assumes the behaviour of the system is a deterministic process (it is not in the N-dimension subspace) chemical systems are not mechanically isolated (5) System Biology - Deterministic & Stochastic Dynamical Systems 16

17 Stochastic Model - Probabilistic Formulation Question: When will the next molecular reaction occur and what type of reaction? Fundamental premise: Reaction µ will occur in [t, t + dt] interval given X (t), with probability a µ (x)dt = f µ c µ dt where c - constant, f - combinatorial function of X P(τ, µ)dτ is the probability that a reaction of type µ occurs in the interval [t + τ, t + τ + dτ]. P(τ, µ) = P 0 (τ)a µ dτ (6) where P 0 (τ) is the probability of NO reaction occurring during [t, t + τ] System Biology - Deterministic & Stochastic Dynamical Systems 17

18 Stochastic Model - Probabilistic Formulation The probability of NO reaction to occur in the interval [t, t + τ + dτ] is P 0 (τ + dτ) = P 0 (τ) (1 µ a µ dτ) where 1 µ a µdτ is the probability of NO reaction to occur in the interval dτ P 0 (τ + dτ) P 0 (τ) dτ with the solution = P 0 (τ) µ a µ dp 0(τ) dτ = P 0 (τ) µ a µ P 0 (τ) = e ( µ a µτ) = e ( a 0τ) (7) where a 0 = µ a µτ System Biology - Deterministic & Stochastic Dynamical Systems 18

19 Stochastic Model - Probabilistic Formulation From Equations 6 and 7 we obtain P(τ, µ) = a µ e a 0τ (8) P(τ, µ) can be written as with P(τ, µ) = P 1 (τ) P 2 (µ) (9) P 1 (τ) = a 0 e ( a 0τ) (10) P 2 (µ) = a µ a 0 (11) System Biology - Deterministic & Stochastic Dynamical Systems 19

20 Stochastic Model - Simulation Algorithm Draw two random numbers r 1, r 2 and take in τ = 1 ln( 1 ) a 0 r 1 µ µ = smallest a i > r 2 a 0 (12) i=1 0. Initialise t = t 0 and X = x Evaluate a µ and their sum a 0 given the system is in state X at time t. 2. Generate τ and µ using equation Update the system t = t + τ, X = X + x µ 4. Record X, t. Return to step 1, or else end simulation. System Biology - Deterministic & Stochastic Dynamical Systems 20

21 Connection with deterministic model A + B k µ 2A k µ = c µ X A X B X A X B X A X B = X A X B k µ c µ a µ = c µ X A X B (13) Formulas of deterministic chemical kinetics are approximate consequences of the formulas of stochastic chemical kinetics Gillespie System Biology - Deterministic & Stochastic Dynamical Systems 21

22 Project2: Simple Decay Example A c Ø (14) For the stochastic model a = cx A For the deterministic model dx A dt = cx A, with the solution X A = X 0 e ct Deterministic Gillespies Simulation X(t) time System Biology - Deterministic & Stochastic Dynamical Systems 22

23 Project2: Bistable Switch Example du dt dv dt = = α V β U α U γ V (15) U(0) = 6, V (0) = u v 200 u v Expression 100 Expression Time Time System Biology - Deterministic & Stochastic Dynamical Systems 23

24 Project2: Bistable Switch Example U(0) = 1, V (0) = u v u v 8 20 Expression 6 4 Expression Time Time System Biology - Deterministic & Stochastic Dynamical Systems 24

25 Project2: Bistable Switch Example U(0) = 1, V (0) = u v u v Expression Expression Time Time System Biology - Deterministic & Stochastic Dynamical Systems 25

26 Lotka Reactions System Biology - Deterministic & Stochastic Dynamical Systems 26

27 Stem Cells System Biology - Deterministic & Stochastic Dynamical Systems 27

28 Landscape Metaphor System Biology - Deterministic & Stochastic Dynamical Systems 28

29 Landscape Metaphor System Biology - Deterministic & Stochastic Dynamical Systems 29

30 Cell Reprogramming System Biology - Deterministic & Stochastic Dynamical Systems 30

31 Cell Reprogramming System Biology - Deterministic & Stochastic Dynamical Systems 31

32 Stem Cell Gene Network System Biology - Deterministic & Stochastic Dynamical Systems 32

33 LIF BMP4 Stem Cell Gene Regulatory Network Model d[n] dt d[os] dt d[fgf ] dt d[g] dt = k 0 [OS](c 0 + c 1 [N] 2 + k 0 [OS] + c 2 LIF ) (1 + (k 0 [OS](c 1 [N] 2 + k 0 [OS] + c 2 LIF + c 3 [FGF ] 2 )) + c 4 [OS][G] 2 ) γ[n], = α + = = (e 0 + e 1 [OS]) (1 + e 1 [OS] + e 2 [G] 2 γ[os], (16) ) (a 0 + a 1 [OS]) γ[fgf ], (1 + a 1 [OS] + a 2 I 3 ) (b 0 + b 1 [G] 2 + b 3 [OS]) (1 + b 1 [G] 2 + b 2 [N] 2 + b 3 [OS]) γ[g], System Biology - Deterministic & Stochastic Dynamical Systems 33

34 LIF BMP4 Project2: Stem Cell Network Example 150 OCT4 SOX2 NANOG LIF+BMP4 150 OCT4 SOX2 NANOG 2i Concentration Time x Time x 10 4 System Biology - Deterministic & Stochastic Dynamical Systems 34

35 Stochastic Simulation Results - Distributions Nanog Oct4 Sox2 LIF+BMP Nanog Oct4 Sox2 2i/3i Density Concentration Concentration System Biology - Deterministic & Stochastic Dynamical Systems 35

36 Transcription Factors Heterogeneity and Ground State Experimental Data Wray et al. Biochemical Transactions (2010). System Biology - Deterministic & Stochastic Dynamical Systems 36

37 Simplified Gene Regulatory Network Topology Young R.A. Cell(2011) Costa et al. Nature(2014) Koh et al. Cell Stem Cell(2011) System Biology - Deterministic & Stochastic Dynamical Systems 37

38 Fast Complex Formation [N free ] + [T free ] [N T ] K d = [N free] [T free ] [N T ] [N total ] = [N free ] + [N T ] [T total ] = [T free ] + [N T ] [N T ] = K d + [N total ] + [T total ] 2 ( ) 2 K d + [N total ] + [T total ] [N total] [T total] 2 System Biology - Deterministic & Stochastic Dynamical Systems 38

39 Slow Gene Regulation [N total ] t [O total ] t [T total ] t [O total ] = N over + LIF + p N K O [N total ] 1 + [O total] K O = [O total ] ( [N T ] ) n K O over + LIF + p O O K 1 + [O NT ( total] [N T ] ) [O total ] n 1 + K O K NT [O total ] ( [N T ] = T over + p T K O 1 + [O total] K O 1 + ) n K NT ( [N T ] ) n K NT [T total ] System Biology - Deterministic & Stochastic Dynamical Systems 39

40 Reprogramming Simulation Results 1 OCT4 NANOG TET1 Expression Level Over-expression 0 Oct4 ON Oct4 OFF Nanog ON Nanog OFF Tet1 ON Tet1 OFF System Biology - Deterministic & Stochastic Dynamical Systems 40

41 Promoter CpG sites methylation and demethylation model - Only For SUPER VG!!! dm dt = σ u m2 κ u 2 m + µ u β m du dt = κ u2 m σ u m 2 µ u + β m select two random CpG sites. If both are in m, then select another CpG site and set its state to m with a probability σ. If both are in u, select another CpG site and set its state to u with a probability κ. select a random CpG site and change it to u with a probability µ if the CpG site is in state m or set it to m with a probability β if the CpG site is in state u. System Biology - Deterministic & Stochastic Dynamical Systems 41

42 The Cell Hill Formalism Shea-Ackers Formalism Complex Formation Gillespie Algorithm Examples DNA methylation model results a b. System Biology - Deterministic & Stochastic Dynamical Systems

43 Double Layer model results Oct4 Nanog Tet1 Expression Level Methylated Unmethylated 0 Over-Expression Oct4 ON Oct4 OFF Nanog ON Nanog OFF Tet1 ON Tet1 OFF System Biology - Deterministic & Stochastic Dynamical Systems 43

44 Project2- The base for passing or good 1. Implement the deterministic and the stochastic (Gillespie algorithm) model for a simple decay. 2. Run simulation of the two models (reproduce the results shown in slide 22). 3. Implement the deterministic and the stochastic (Gillespie algorithm) model for the bistable switch. 4. Conduct simulations for both models with initial conditions specified on slides (23, 24, 25) and with parameters specified in Gardner et al. (Reproduce the results shown in slides 23, 24, 25.) 5. Reproduce figure 8c from Gillespie Help: implement Gillespie algorithm for Lotka reactions equations (38, 39). Plot X vs. Y. Use the values of parameters specified in figure caption. (Slide 26) System Biology - Deterministic & Stochastic Dynamical Systems 44

45 Project2- for VG and Super VG 6. Implement the stochastic model of the Stem Cell Network, reproduce the results shown in slides 34 and 35. Use the parameters specified in the table from Chickarmane et al. 7. Implement the deterministic model for reprogramming with protein complex formation. Reproduce the results in slide 40. The Parameters and models details can be found in Olariu et al. 8. Implement the double layer model for reprogramming. Reproduce the results shown in slide 43. Details in Olariu et al. plus supplementary material. System Biology - Deterministic & Stochastic Dynamical Systems 45