computational methods for systems and synthetic...

21/10/2013 François Fages C2-19 MPRI 1

Computational Methods for

Systems and Synthetic Biology

François Fages

The French National Institute for Research in

Computer Science and Control

INRIA Paris-Rocquencourt

Constraint Programming Group

http://contraintes.inria.fr/


Overview of the Course

1. Rule-based modeling of biochemical systems

• Syntax: molecules, reactions, regulations, SBML/SBGN Biocham notations

• Semantics: Boolean, Differential and Stochastic interpretations of reactions

• Influence graph abstraction

• Static analyses: consistency, influence graph circuits, protein functions,…

• Examples in cell signaling, gene expression, virus infection, cell cycle

2. Temporal Logic based formalization of biological properties

• Qualitative model-checking in propositional Computation Tree Logic CTL

• Quantitative model-checking in Linear Time Logic LTL(R)

• Parameter search in high dimension w.r.t. LTL(R) specifications

• Robustness and sensitivity analyses w.r.t. LTL(R) specifications


Biochemical Kinetics

• Study the concentration of chemical substances in a biological system as

a function of time.

• Differential and stochastic semantics of Biocham

Molecules: A1 ,…, Am

|A|=Number of molecules A

[A]=Concentration of A in the solution: [A] = |A| / Volume ML-1

Solution: S, S’, …

multiset of molecules

formal linear expression with stoichiometric coefficients: c1 *A1 +…+ cn * An


Mass Action Law Kinetics

Assumption: each molecule moves independently of other molecules in a random

walk (diffusion, dilute solutions, low concentration)

Law: The number of interactions of A with B is proportional to the number of A

and B molecules present in the solution

The rate of a reaction A + B k C is k*A*B for some reaction constant k

The time evolution of concentrations obeys the ordinary differential equation

dC/dt = k A B

dA/dt = . . .


Mass Action Law Kinetics

Assumption: each molecule moves independently of other molecules in a random

walk (diffusion, dilute solutions, low concentration)

Law: The number of interactions of A with B is proportional to the number of A

and B molecules present in the solution

The rate of a reaction A + B k C is k*A*B for some reaction constant k

The time evolution of concentrations obeys the ordinary differential equation

dC/dt = k A B

dA/dt = -k A B

dB/dt = -k A B


Differential Semantics of Reaction Rules

• To a set of reaction rules with kinetic expressions ei

{ ei for Si => S’i }i=1,…,n

one associates the Ordinary Differential Equation (ODE) over {A1 ,…, Ak}

dAk/dt = Σni=1 ( ri(Ak) - li(Ak) ) * ei

where ri(A) is the stoichiometric coefficient of A in Si

li(A) is the stoichiometric coefficient of A in S’i .


Differential Semantics of Reaction Rules

• To a set of reaction rules with kinetic expressions ei

{ ei for Si => S’I }i=1,…,n

one associates the Ordinary Differential Equation (ODE) over {A1 ,…, Ak}

dAk/dt = Σni=1 ( ri(Ak) - li(Ak) ) * ei

where ri(A) is the stoichiometric coefficient of A in Si

li(A) is the stoichiometric coefficient of A in S’i .

• Inverse problem: load_ode(file)


Numerical Integration Methods

System dX/dt = f(X).

Initial conditions X0

Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, …

and compute a trace

(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…






and compute a trace


• Euler’s method: ti+1=ti+ Δt

Xi+1=Xi+f(Xi)*Δt

error estimation E(Xi+1)=|f(Xi)-f(Xi+1)|*Δt






and compute a trace



Xi+1=Xi+f(Xi)*Δt


• Runge-Kutta’s method: intermediate computations at Δt/2






and compute a trace



Xi+1=Xi+f(Xi)*Δt



• Adaptive step method: Δti+1= Δti/2 while E>Emax, otherwise Δti+1= 2*Δti






and compute a trace



Xi+1=Xi+f(Xi)*Δt



• Adaptive step method: Δti+1= Δti/2 while E>Emax, otherwise Δti+1= 2*Δti

• Rosenbrock’s stiff method: solve Xi+1=Xi+f(Xi+1)*Δt by formal differentiation


Signal Reception on the Membrane

present(L,0.5). present(RTK,0.01).

absent(L-RTK). absent(S).

parameter(k1,1). parameter(k2,0.1).


(k1*[L]*[RTK], k2*[L-RTK]) for L+RTK <=> L-RTK.

(k3*[L-RTK]^2, k4*[S]) for 2*(L-RTK) <=> S.

dS/dT = …









dS/dT = k3*[L-RTK]^2 – k4*[S]

d(L-RTK)/dT = …









dS/dT = k3*[L-RTK]^2 – k4*[S]

d(L-RTK)/dT = k1*[L]*[RTK] + 2*k4*[S} –k2*[L-RTK] – 2*k3*[L-RTK]^2


Compositionality of Reaction Rules

The union of two sets of reaction rules is a set of reaction rules… hence

rule-based models can be composed to form complex reaction models by

set union (or by multiset union with addition of the kinetics [Plotkin 05])






Pb with set union: equality on rules (Biocham congruence is not enough)

Pb with multiset union: duplicated rules get double rates






Pb with set union: equality on rules (Biocham congruence is not enough)

Pb with multiset union: duplicated rules get double rates

In a rule-based modular model:

• All molecular species can serve as interface variables

– controlled by other modules (endogeneous variables)

– controlled by external laws (exogeneous variables).

• Privacy is obtained by membranes (cell compartments, signalling vesicles)

• Reactions must be sufficiently decomposed to provide module interfaces

• Sufficiently general kinetics expression to be applicable in different contexts

– possibly functions of pH, temperature,…


Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S k1 C k2 E+P

E+S km1 C

• Mass action law kinetics ODE:

dE/dt =…




E+S k1 C k2 E+P

E+S km1 C


dE/dt = -k1ES+(k2+km1)C

dS/dt = -k1ES+km1C

dC/dt = k1ES-(k2+km1)C

dP/dt = k2C

with two conservation laws (Σni=1 Mi = constant if Σn

i=1 dMi/dt = 0):




E+S k1 C k2 E+P

E+S km1 C


dE/dt = -k1ES+(k2+km1)C

dS/dt = -k1ES+km1C

dC/dt = k1ES-(k2+km1)C

dP/dt = k2C

with two conservation laws (Σni=1 Mi = constant if Σn

i=1 dMi/dt = 0):

E+C=constant, S+C+P=constant,

• Assuming C0=P0=0, we get E=E0-C and S0=S+C+P

dS/dt = -k1(E0-C)S+km1C dC/dt = k1(E0-C)S-(k2+km1)C


Multi-Scale Phenomenon Hydrolysis of benzoyl-L-arginine ethyl ester by trypsin

present(En,1e-8). present(S,1e-5). absent(C). absent(P). En << S

parameter(k1,4e6). parameter(km1,25). parameter(k2,15). k1 >> k2 (k1*[En]*[S], km1*[C]) for En+S <=> C.

k2*[C] for C => En+P.






Complex formation 5e-9 in 0.1s






Complex formation 5e-9 in 0.1s Product formation 1e-5 in 1000s


Quasi-Steady State Approximation

Assume dC/dt=0=dE/dt

(E<<S, k1 >>k2)

we get C = E0S/(Km+S)

where Km=(k2+km1)/k1

Leonor Michaelis Maud Menten 1913

dP/dt = -dS/dt = VmS / (Km+S)

where Vm= k2E0


(E<<S, k1 >>k2)




where Vm= k2E0 maximum velocity at saturing substrate concentration




(E<<S, k1 >>k2)



substrate concentration with half maximum velocity






(E<<S, k1 >>k2)



substrate concentration with half maximum velocity



Michaelis-Menten kinetics: VmS /(Km+S) for S => P

C is eliminated, E is supposed constant and acting on only one substrate

the enzyme E is no longer a system’s variable

should not write VmS /(Km+S) for S =[E]=> P




Michaelis Menten Kinetics

parameter(E0,1e-8).

macro(Vm, k2*E0).

macro(Km, (km1+k2)/k1).

MM(Vm,Km) for S ==> P.

or equivalently

macro(Kf, Vm*[S]/(Km+[S])).

Kf for S ==> P.

ODE vs Stochastic Approaches

num

ber

of dis

cre

te s

tate

pro

babili

ties

chemical master equation

number of moments for describing the joint distribution of fluid variables

infinitely many moments

2 1 deterministic

limit/ ODE

approach

linear noise approximation

moment closure/MM

hybrid fluid models

(Fluid Stochastic

Petri Nets, Stochastic

Hybrid Automata,…)

truncation approaches (FSP, FAU, …)

After Verena Wolf,

U. Saarbrucken

Moment Equations (1)

Start from master equation:

x is vector of molecule numbers

is propensity of jth reaction

is change vector of jth reaction

is probability of x at time t


Consider time evolution of mean:

plug master equation in and simplify ...

involves higher moments if jth reaction is at least bimolecular, e.g.

next: replace by a Taylor series about mean After Verena Wolf,

U. Saarbrucken


time evolution of the mean:

Taylor series (one-dimensional case) about the mean:

and thus

=0 in general: (co-)variances appear here

and give more detailed description of

time evolution of the mean

if law of mass action and at most

bimolecular reactions,

terms of order >2 are zero

order 1 approx

After Verena Wolf,

U. Saarbrucken

Moment Equations – Example 1

Example: order 1 approximation:

After Verena Wolf,

U. Saarbrucken

Moment Closure Example 1

Example:

order-2 approximation:

exact time evolution of means:

=> approximate time evolution of covariance

After Verena Wolf,

U. Saarbrucken


Close after 2nd

moment => approximation because we set all higher centered moments to zero

Equations for rate functions that are at most quadratic:

Integrating this is as easy as implementing the SSA!!

There are many examples where means + covariances give a very good approximation since distributions are often similar to multivariate normal distribution (even when populations are small)

After Verena Wolf,

U. Saarbrucken


time evolution of means (including approximation of the covariances) and covariances

After Verena Wolf,

U. Saarbrucken


Species

Proteins: X1, X2

Promoter state: DNA, DNA.X1, DNA.X2

Exclusive Switch (Loinger et al., Phy. Rev. E, 2007)

Reactions

DNA ⟶ DNA+ Xi

Xi ⟶ 0

DNA+Xi ⟷ DNA.Xi

DNA.Xi ⟶ DNA.Xi + Xi

After Verena Wolf,

U. Saarbrucken

Moment Closure Example 2 Exclusive Switch (Loinger et al., Phy. Rev. E, 2007)

0 50 1000

50

100

P1

P2

0 50 1000

50

100

P1

P2

X1

X2

X1

X1

X2

⟵ low rate for binding to the promoter

high rate for binding to the promoter ⟶


X1

low binding rate

After Verena Wolf,

U. Saarbrucken


X1

high binding rate

order 4

(normal

pdf uses only mean and covariance matrix)

Exclusive Switch (Loinger et al., Phy. Rev. E, 2007)


X1

X2

X2



how good is moment

closure?

means

means

order 2 approximation


X1



(co)variances

(co)variances


Exclusive Switch:

X1

Covariances

(high binding rate)

order 2

order 4


Lotka-Volterra Oscillator

0.3*[RA] for _ =[RA]=> RA. 0.3*[RA]*[RB] for RA =[RB]=> RB.

0.15*[RB] for RB => RP. present(RA,0.5). present(RB,0.5). absent(RP).

Moment Closure Lotka-Volterra model:

probability of extinction (here: of Y)

cannot be adequately represented no accurate approximation

X1

0 3 6 9 12 15 18 21 24 27 300

100

200

300

400

time

po

pu

latio

n s

ize

prey

predator

(a) Sustained oscillations of the deterministic Lotka-Volterra model.

0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

time

exp

ect

ed

po

pu

latio

ns

prey

predator

(b) Expected populations in the stochastic Lotka-Volterra model.

Fig. 1. ODE solution and stochastic solution of the Lotka-Volterra model where α = γ = 1, β = 0.01, and x0 = y0 = 20.

We extend these methods in such a way that the cheap but

inaccurate method of moments becomes more accurate and the

expensive but accurate numerical integration becomes faster.

In Section V, we propose a stochastic hybrid approach that

dynamically switches between a moment-based representation

and a distribution-based representation as well as having the

ability to simultaneously use a combination of both. We

implemented all methods in C++ and performed extensive

simulations of the stochastic Lotka-Volterra model in order

to provide comparisons in terms of accuracy and run time.

Numerical experiments on a number of problems indicate that

the stochastic hybrid approach is overall the best method. It

is at least an order faster than the numerical integration of the

master equation and yields no more than a 5 % relative error

in the first moment of the transient distribution.

II. STOCHASTIC LOTKA-VOLTERRA MODEL

We describe the transitions of the stochastic Lotka-Volterra

model in a guarded command style [1, 17]. A guarded com-

mand has the form guard |- rate -> update; and it

describes a set of transitions in the underlying Markov process.

The guard is a Boolean predicate that determines in which

states the transition is possible, the rate is the real-valued

function that evaluated in the current state gives the positive

transition rate in the Markov process, and update is the

function that calculates the successor state of the transition. Let

x and y be the positive integers that represent the populations

of prey and predator, respectively. The three different possible

transitions are given as follows:

x > 0 |- α · x -> x := x + 1;

x > 0, y > 0 |- β · x · y -> x := x − 1, y := y + 1;

y > 0 |- γ · y -> y := y − 1;

The above guarded command model specifies the elements

of the infinitesimal generator matrix Q of a Markov process

{ (X (t), Y (t)) , t ≥ 0} which takes states (x, y) in N2. More

precisely, if for a pair (x, y) the guard is true, then the element

that belongs to (x, y) and the update of (x, y) is equal to the

rate. E.g. the row of state (2, 2) contains three positive entries.

One for the transition to state (3, 2) at rate 2α, one for the

transition to state (1, 3) at rate 4β, and one for the transition to

state (2, 1) at rate 2γ . The diagonal element is then given by

the negative sum of the off-diagonal elements. It is easy to see

that { (X (t), Y (t)) , t ≥ 0} is a regular Markov process [2].

We remark that the outflow rate from state (x, y) with x, y > 0

is (αx + γy + βxy), where α, γ,β > 0, and this rate increases

unboundedly with increasing values of x and y.

Let us fix the initial state of the system as (x0, y0). Then

the transient probability distribution is given by the master

equationddt

p( t ) (x, y) = M (p( t ) (x, y)), (2)

where p( t ) (x, y) = P{ X (t) = x, Y (t) = y|X (0) = x0,

Y(0) = y0} . The master operator M is defined for any real-

valued function g : N2 → R such that M (g) is the function

that maps a state (x, y) to the value1

M (g(x, y)) = α(x − 1)g(x − 1, y)

+ β(x + 1)(y − 1)g(x + 1, y − 1)

+ γ(y + 1)g(x, y + 1)

− (αx + βxy + γy)g(x, y),

(3)

where α,β, γ are the rate constants as defined before and

x, y > 0. If x = 0 and/or y = 0, then the terms involving

g(x − 1, y) and/or g(x + 1, y − 1) are removed. The ordinary

first-order differential equation in (2) is a direct consequence

of the Kolmogorov forward equation and describes the change

of the probability distribution as the difference between inflow

of probability from direct predecessors (first three terms) and

outflow of probability in state (x, y) (last term).

III. DIRECT NUMERICAL APPROXIMATION

Since no analytical solution is known for Eq. (2) and the

number of equations is infinite in two dimensions, a numerical

solution is only possible if appropriate bounds for the variables

x and y are found.

1We assume that all terms with a negative argument are zero(e.g. α (x − 1)g(x − 1, y) = 0 if x < 1).

Reactions

X ⟶ 2X

X+Y ⟶ 2Y

Y ⟶ 0

order 1 closure true expectations


Enzymatic Competitive Inhibition present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[C]*[I] for C+I => CI.



Isosteric Inhibition present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[E]*[I] for E+I => EI.

Complex formation 2.5e-9 in 0.4s Product formation 2.5e-9 in 1000s


Allosteric Inhibition/Activation

(i*[E]*[I],im*[EI]) for E+I <=> EI. parameter(i,1e7). parameter(im,10).

(i1*[EI]*[S],im1*[CI]) for EI+S <=> CI. parameter(i1,5e6). parameter(im1,5).

i2*[CI] for CI => EI+P. parameter(i2,2).



Hill Kinetics for Allosteric Enzymatic Reactions

Dimer enzyme with two promoters:

E+S 2*k1 C1 k2 E+P C1+S k’1 C2 2*k’2 C1+P

E+S k-1 C1 C1+S 2*k’-1 C2

Let Km=(k-1+k2)/k1 and K’m=(k’-1+k’2)/k’1

Non-cooperative if K’m =Km

Michaelis-Menten rate: VmS / (Km+S) where Vm=2*k2*E0.

hyperbolic velocity vs substrate concentration

Cooperative if K’m >Km

Hill equation rate: VmS2 / (Km*K’m+S2) order 2

where Vm=2*k2*E0.

sigmoid velocity vs substrate concentration


Implementing Gene Regulatory Networks

• Gene a, product Pa, promotion factor PFa (same for gene b)

#a + PFa => #a-PFa. #b + PFb => #b-PFb.

_ =[#a-PFa]=> Pa. _ =[#b-PFb]=> Pb.





_ =[#a-PFa]=> Pa. _ =[#b-PFb]=> Pb.

• a activates b “a b” (the product of a is the promotion factor of b)





_ =[#a-PFa]=> Pa. _ =[#b-PFb]=> Pb.


Pa = PFb

• a inhibits b “a --| b” (first implementation by transcriptional inhibition)




#a + PFa <=> #a-PFa. #b + PFb <=> #b-PFb.

_ =[#a-PFa]=> Pa. _ =[#b-PFb]=> Pb.


Pa = PFb


#b + Pa <=> #b-Pa.

• a inhibits b “a --| b” (second implementation by promotion factor inhibition)




#a + PFa <=> #a-PFa. #b + PFb <=> #b-PFb.

_ =[#a-PFa]=> Pa. _ =[#b-PFb]=> Pb.


Pa = PFb


#b + Pa <=> #b-Pa.

• a inhibits b “a --| b” (second implementation by promotion factor inhibition)

PFb + Pa <=> PFb-Pa.


Gene Circuit

• a activates a, a activates b, b inhibits a

• First implementation:

#a + Pa <=> #a-Pa.

_ =[#a-Pa]=> Pa.

#b + Pa <=> #b-Pa.

_ =[#b-Pa]=> Pb.

#a + Pb <=> #a-Pb.


Hybrid (Continuous-Discrete) Dynamics

Gene X activates gene Y but above some threshold gene Y

inhibits X.

absent(X). absent(Y).

if [Y] < 0.8 then 0.01 for _ => X.

0.01*[X] for X => X + Y.

0.02*[X] for X => _.


T7 Bacteriophage Infection Process

Retrovirus:

• Template nucleic acids (RNA): tem (initial condition of low infection)

• Genomic nucleic acids (DNA): gen

• Structural proteins: struc

MA(c1) for gen=>tem.

MA(c2) for tem=>_.

MA(c3) for tem=>tem+gen.

MA(c4) for gen+struc=>virus.

MA(c5) for tem=>tem+struc.

MA(c6) for struc=>_.

parameter(c1,0.025).

parameter(c2,0.25).

parameter(c3,1.0).

parameter(c4,0.0000075).

parameter(c5,1000).

parameter(c6,1.99).


T7 Bacteriophage ODE semantics

MA(c1) for gen=>tem.

MA(c2) for tem=>_.

MA(c3) for tem=>tem+gen.

MA(c4) for gen+struc=>virus.

MA(c5) for tem=>tem+struc.

MA(c6) for struc=>_.

[tem] =1

d[virus]/dt=c4*[gen]*[struc]

d[tem]/dt=c1*[gen]-c2*[tem]

d[struc]/dt=c5*[tem]-c6*[struc]-c4*[gen]*[struc]

d[gen]/dt=c3*[tem]-c4*[gen]*[struc]-c1*[gen]


T7 Bacteriophage ODE semantics

2 steady states dX/dt=0 :

1) tem=gen=struc=0

2) tem=20 gen=200 struc=10000

cyclic behavior

Minimal T-invariants in Petri nets

I*V=0 (incidence stoichiometric matrix)

Elementary Modes of flux distribution

in metabolic networks

tem = - (c1*c6)/(- (c2*c5)-c1*c5+c3*c5)*(c2*c4/(- (c1*c3)+c2*c3))^ - 1

gen = ((c2*c6/(- (c3*c5)+c2*c5+c1*c5))^ - 1*(c2*c4/(- (c1*c3)+c2*c3)))^ - 1

struc = (c2*c4/(- (c1*c3)+c2*c3))^ - 1


T7 Bacteriophage Stochastic semantics

Similar viral explosion Recovery

Two qualitatively different behaviors:


T7 Bacteriophage Boolean Semantics

Querying all possible behaviors in Computation Tree Logic

(symbolic model-checking NuSMV)

biocham: nusmv( EF ( !(gen) & !(tem) & !(struc) & !(virus))).

true

biocham: why.

tem is present

2 tem=>_.

tem is absent

Query time: 0.00 s


T7 Bacteriophage Boolean Semantics

biocham: nusmv( EF ( virus & EF ( !(gen) & !(tem) & !(struc)) )).

true

biocham: why.

tem is present

5 tem=>struc+tem.

struc is present

3 tem=>gen+tem.

gen is present

4 gen+struc=>virus.

gen is absent

struc is absent

virus is present

2 tem=>_.

tem is absent


Aut. Generation of Temporal Logic

Properties

Ei(reachable(gen))

Ei(reachable(!(gen)))

Ei(steady(!(gen)))

Ai(checkpoint(tem,gen)))

Ei(reachable(tem))

Ei(reachable(!(tem)))

Ei(steady(tem))

Ai(checkpoint(gen,tem)))

Ei(reachable(struc))

Ei(reachable(!(struc)))

Ei(steady(!(struc)))

Ai(checkpoint(tem,struc)))

Ei(reachable(virus))

Ei(reachable(!(virus)))

Ei(steady(!(virus)))

Ai(checkpoint(gen,virus)))

Ai(checkpoint(gen,!(virus))))

Ai(checkpoint(tem,!(virus))))

Ai(checkpoint(struc,virus)))

Time: 0.06 s

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