lecture 5 the futurebortolu/files/didattica/sissacourse... · 2007-01-31 · lecture 5 the future...

Introduction sCCP Bio-Modeling S-Systems and Diff. Eq. Automata and Temporal Logic

Lecture 5The future

Luca Bortolussi1 Alberto Policriti2

1Dipartimento di Matematica ed InformaticaUniversità degli studi di TriesteVia Valerio 12/a, 34100 Trieste.

[email protected] di Matematica ed Informatica

Università degli studi di UdineVia delle Scienze 206, 33100 Udine.

[email protected]

SISSA, January 2007


Material from:

... on going work.


Systems Biology, Computer Science, and Communication .

Formal tools for specifying Communicating andConcurrent systems have been used as a paradigm for:

1 computability (classic - Milner Calculus of CommunicatingSystems)

2 biological modeling (rather new - π-calculus Shapiro, Priami,Cardelli, ...)

Our proposal:an attempt to establish a closer connection betweencommunication-based and differential equations modeling techniques

Our tool: Stochastic Concurrent Constraint Programming(sCCP) as a link between:

Discrete and Stochastic (Gillespie algorithm)

Continuous and Deterministic modeling (differential equations)


Biological Systems and Stochastic Process Algebras

Molecule ProcessInteraction capability Channel

Interaction CommunicationModification State change

(of cellular components) (state transition systems)

A. Regev and E. Shapiro Nature, 2002.

ProsSimple Language

Compositionality

ConsHard to encodegeneral information

Lackingcomputationalextensibility


The picture


Concurrent Constraint Programming

Constraint StoreIn this process algebra, the main objects are agents and constraints

Constraints are formulae over an interpreted first order language (i.e.X = 10, Y > X − 3).

Constraints can be added to a "pot", called the constraint store, but cannever be removed.

AgentsAgents can perform two basic operations on this store:

Add a constraint (tell instruction)

Ask if a certain relation is entailed by the current configuration (askinstruction)


Continuous Time Markov Chains

A Continuous Time Markov Chain (CTMC) is a direct graph with edgeslabeled by a real number, called the rate of the transition (representing thespeed or the frequency at which the transition occurs).

In each state, we select the next stateaccording to a probability distributionobtained normalizing rates (from S to S1

with prob. r1r1+r2

).

The time spent in a state is given by anexponentially distributed random variable,with rate given by the sum of outgoingtransitions from the actual node (r1 + r2).


Mass Action Kinetics in sCCP

ExampleR1 + . . . + Rn →k P1 + . . . + Pm

reaction(k , [R1, . . . , Rn], [P1, . . . , Pm]) : −

askrMA(k ,R1,...,Rn)

(n∧

i=1

(Ri > 0)

).

(‖n

i=i tell∞(Ri = Ri − 1) ‖mj=1 tell∞(Pj = Pj + 1)

).

reaction(k , [R1, . . . , Rn], [P1, . . . , Pm])

where rMA(k , X1, . . . , Xn) = k · X1 · · ·Xn.


Biochemical Arrows to sCCP processes

R1 + . . . + Rn →k P1 + . . . + Pm

reaction(k, [R1, . . . , Rn ], [P1, . . . , Pm ]) : −askrMA(k,R1,...,Rn)

�Vni=1(Ri > 0)

�.�

‖ni=i tell∞(Ri = Ri − 1) ‖m

j=1 tell∞(Pj = Pj + 1)�.

reaction(k, [R1, . . . , Rn ], [P1, . . . , Pm ])

R1 + . . . + Rn k1k2

P1 + . . . + Pmreaction(k1, [R1, . . . , Rn ], [P1, . . . , Pm ]) ‖reaction(k2, [P1, . . . , Pm ], [R1, . . . , Rn ])

S 7→EK ,V0

P

mm_reaction(K , V0, S, P) : −askrMM (K ,V0,S)(S > 0).

(tell∞(S = S − 1) ‖ tell∞(P = P + 1)) .mm_reaction(K , V0, S, P)

S 7→EK ,V0,h P

hill_reaction(K , V0, h, S, P) : −askrHill (K ,V0,h,S)(S > 0).

(tell∞(S = S − h) ‖ tell∞(P = P + h)) .Hill _reaction(K , V0, h, S, P)

where rMA(k, X1, . . . , Xn) = k · X1 · · · Xn ; rMM (K , V0, S) =V0S

S + K; rHill (k, V0, h, S) =

V0Sh

Sh + K h


Enzymatic reaction

S + E k1k−1

ES →k2 P + E

Mass Action Equationsd [ES]

dt = k1[S][E ]− k2[ES]− k−1[ES]d [E ]dt = −k1[S][E ] + k2[ES] + k−1[ES]

d [S]dt = −k1[S][E ]

d [P]dt = k2[ES]

Mass Action Kineticsenz_reaction(k1, k−1, k2, S, E, ES, P) :-

reaction(k1, [S, E ], [ES]) ‖reaction(k−1, [ES], [E, S]) ‖reaction(k2, [ES], [E, P])

Michaelis-Menten Equationsd [P]dt =

V0SS+K

V0 = k2[E0]

K =k2+k−1

k1

Michaelis-Menten Kinetics

mm_reaction

k2 + k−1

k1, k2 · E, S, P

!


Enzymatic reaction

S + E k1k−1

ES →k2 P + E

Mass Action Equationsd [ES]

dt = k1[S][E ]− k2[ES]− k−1[ES]d [E ]dt = −k1[S][E ] + k2[ES] + k−1[ES]

d [S]dt = −k1[S][E ]

d [P]dt = k2[ES]

Michaelis-Menten Kinetics

mm_reaction

k2 + k−1

k1, k2 · E, S, P

!


MAP-Kinase cascade (C.F. Huang and J.T. Ferrel. PNAS, Biochemistry, 1996.)

enz_reaction(ka, kd , kr , KKK , E1, KKKE1, KKKS) ‖ enz_reaction(ka, kd , kr , KKKS, E2, KKKSE2, KKK ) ‖enz_reaction(ka, kd , kr , KK , KKKS, KKKKKS, KKP) ‖ enz_reaction(ka, kd , kr , KKP, KKP1, KKPKKP1, KK ) ‖

enz_reaction(ka, kd , kr , KKP, KKKS, KKPKKKS, KKPP) ‖ enz_reaction(ka, kd , kr , KP, KP1, KPKP1, K ) ‖enz_reaction(ka, kd , kr , K , KKPP, KKKPP, KP) ‖ enz_reaction(ka, kd , kr , KKPP, KKP1, KKPPKKP1, KKP) ‖

enz_reaction(ka, kd , kr , KP, KKPP, KPKKPP, KPP) ‖ enz_reaction(ka, kd , kr , KPP, KP1, KPPKP1, KP)


The gene machine instruction set

null_gate(kp, X) : −tellkp (X = X + 1).null_gate(kp, X)

pos_gate(kp, ke, kf , X , Y ) : −tellkp (X = X + 1).pos_gate(kp, ke, kf , X , Y )

+askr(ke,Y )(true).tellke (X = X + 1).pos_gate(kp, ke, kf , X , Y )

neg_gate(kp, ki , kd , X , Y ) : −tellkp (X = X + 1).neg_gate(kp, ki , kd , X , Y )

+askr(ki ,Y )(true).askkd(true).neg_gate(kp, ki , kd , X , Y )

where r(k, Y ) = k · Y .


Repressilator (M. B. Elowitz and S. Leibler. Nature, 2000.)

neg_gate(0.1, 1, 0.0001, A, C) ‖reaction(0.0001, [A], []) ‖

neg_gate(0.1, 1, 0.0001, B, A) ‖reaction(0.0001, [B], []) ‖

neg_gate(0.1, 1, 0.0001, C, B) ‖reaction(0.0001, [C], [])


Circadian Clock (J. M. G. Vilar, H. Yuan Kueh, N. Barkai, and S. Leibler. PNAS, 2002.)


Circadian Clock

p_gate(αA, α′A, γA, θA, MA, A) ‖p_gate(αR , α′R , γR , θR , MR , A) ‖reaction(βA, [MA], [A]) ‖reaction(δMA, [MA], []) ‖reaction(βR , [MR ], [R]) ‖reaction(δMR , [MR ], []) ‖reaction(γC , [A, R], [AR]) ‖reaction(δA, [AR], [R]) ‖reaction(δA, [A], []) ‖reaction(δR , [R], [])


S-Systems

S-systems describe the dynamical behavior of a biologicalsystem by a set of differential equations over reactants

non-linear, time-invariant, DAE systems;

biologically plausible and expressive;

analytical approximation power.

DefinitionAn S-system’s equation has the form:

Xi = αi

n+mYj=1

Xgijj − βi

n+mYj=1

Xhijj

with αi , βi ≥ 0 called rate constants and gij , hij called kinetic orders.

E. O. Voit. Computational Analysis of Biochemical Systems A Practical Guide for Biochemists andMolecular Biologists. Cambridge University Press, 2000.

M. A. Savageau. Biochemical System Analysis: A Study of Function and Design in Molecular Biology.Addison-Wesley, 1976.


Using S-Systems to determine rate functions

A generic form for S-System equations is

Xi = V +(X1, . . . , Xm+n)| {z }production speed

−V−(X1, . . . , Xm+n)| {z }degradation speed

.

A generic S-System equation has non-linear dependencies on variables.

Xi = αi

n+mYj=1

Xgijj − βi

n+mYj=1

Xhijj

We can use these expression as rates.

Using sCCP, we can associate to each dependent variable an agent subs(Xi): tell(Xi = Xi + σ)�

αiQn+m

j=1 Xgijj

� + tell(Xi = Xi − σ)�βiQn+m

j=1 Xhijj

�!

.subs(Xi).


Encoding S-System’s Repressilator

X1 = α1X−13 − β1X 0.5

1 ,

X2 = α2X−11 − β2X 0.5

2 ,

X3 = α3X−12 − β3X 0.5

3 ,

αi = 0.2, βi = 1.

subs(X1) ::= (tell(X1 = X1 + σ)[α1X−13 ]

+tell(X1 = X1 − σ)[β1X0.51 ]).subs(X1)

subs(X2) ::= (tell(X2 = X2 + σ)[α2X−11 ]

+tell(X2 = X2 − σ)[β2X0.52 ]).subs(X2)

subs(X3) ::= (tell(X3 = X3 + σ)[α3X−12 ]

+tell(X3 = X3 − σ)[β3X0.53 ]).subs(X3)

σ = 1


In progress:Systematic translation and behavioral equivalence theorem

(Probabilistic) Model Checking

Reachability

Reimplementation

L. Bortolussi Constraint-Based Approaches to StochasticDynamics of Biochemical Systems. Ph.D. thesis, Universityof Udine.


ConclusionsA stochastic version of CCP, with functional rates, can helpin “programming” biological (biochemical, genetic)communication networks

sCCP can be used for modeling and checking biologicalsystems, defining libraries for biochemical reactions andgene regulatory networksIt allows a decoupling of

discrete and procedural programming activity (discrete)continuous simulation (functional rates)


Automata for modeling

The general ideaMerge modeling techniques from the mathematical and logicaltraditions using automata as common ground.

Automatastates as tuple of variables’values of the underlying(mathematical) model;

an automated query mechanism/language can beprovided.


The temporal ingredient

Automaton-based models of the temporal evolution of a system

can be built on (heterogeneous) data;

can be queried.

Timed query languageTime dimension built-in the query language: CTL (ACTL∗, LTL,. . . ) language to query the system.


The data

Heterogeneous dataProduced as numerical traces of:

1 in vivo,2 in vitro, and3 in silico ,

experiments.

Quantitative aspects must be considered (e.g. in the querylanguage).


The interaction with the system

given a biological system described by means ofa pathway

a system of differential equations

a set of traces

We want to allow the user to formulate queries about thesystems’s qualitative behavior, e.g.

will the system reach a steady state?

verify:“... the two scaffold proteins axin and APCpromotethe formation of degradation complexes in very differentways. ...”

will concentration of reactant IMP stay over level 0.5?


Computation Tree Logic (CTL) structures

Formally our automata will be:

DefinitionA CTL structure (branching time structure) is M = (S, R, L)such that:

S set of states

R ⊆ S × S binary relation on states

L : S → Pow(AP) labeling function

Capture the intuition about the behavior of reactivesystems;

look like a graph rather than a tree: unfold it!


CTL language

Built using path quantifiers A and E plus the followingoperators:

Next Time: XP ... property P holds in the second state ofthe path;

Eventually: FP ... property P holds at some future state inthe path (in the future);

Always: GP ... property P holds at every state on the path(globally);

Until: PUQ ... property Q holds at every state on the pathand property P holds until then;

Release: PRQ ... property Q holds at every state on thepath up to the first state at which P holds.


Model Checking: automatic and formal validation

A classic in verificationTruth of a CTL-formulae can be checked automatically onCTL-structures.

Model Checking Algorithmassociate a set of labels with each state (i.e., the set ofsub-formulae true at that state);

initialize the set of labels in a given state with atomicpropositions (looking at the input);

proceed inductively on the structural complexity offormulae to extend the set of labels: 7 recursive calls to aprocedure label-graph (P,¬p, p ∧q, AX p, EX p,A(p U q), E(p U q)).


Complexity

Model checking is linear in the size of the model

Theorem (Clarke-Emerson-Sistla, Lichtenstein-Pnueli)Truth of a CTL(LTL)-formula p in a CTL(LTL)-structure (S, R, L)can be established in time O(|p|(|S|+ |R|)) (O(2|p|(|S|+ |R|))).

However ...

State explosion problemThe size of the structure can be huge.


The mathematical modeling machinery: S-Systems

A biological system usually involves

dependent reactants (variables) whose values are affected by thesystem

independent reactants that are not affected by the system

S-systems describe the dynamical behavior of a biologicalsystem by a set of differential equations over reactants

non-linear, time-invariant, DAE systems;

biologically plausible and expressive (see Savageau, Voit, ...);

they are an interchangeable component of the approach

E. O. Voit. Computational Analysis of Biochemical Systems A Practical Guide for Biochemists andMolecular Biologists. Cambridge University Press, 2000.

F. Alvarez-Vasquez, K. J. Sims, L. A. Cowart, Y. Okamoto, E. O. Voit, Y. A. Hannun Simulation and validationof modelled sphingolipid metabolism in Saccharomyces cerevisiae. Nature, vol. 433, 27 January 2005.


S-Systems: formal definition

DefinitionAn S-system is a tuple S = (DV , IV , DE , C) such that:

DV = {X1, . . . , Xn} dependent variables

IV = {Xn+1, . . . , Xn+m} independent variables

DE is a set of differential equations of the form

Xi = αi

n+m∏j=1

Xgij

j − βi

n+m∏j=1

Xhij

j

with αi , βi ≥ 0 called rate constants;

C is a set of algebraic constraints


Traces

The dynamic behavior of an S-system can be simulated bycomputing the (approximate) values of its variables at differenttime instants.

Definition

A system’s trace tr(S, t0, ~X (t0), s, tf ) =〈~g(t0), ~g(t0 + s), . . . , ~g(t0 + (j − 1) ∗ s), ~g(t0 + j ∗ s)〉 is obtainedby fixing:

an initial time t0the initial values ~X (t0) of the system variables

a time step s

a final time tf = to + j ∗ s, (j ≥ 0)


From Traces to Automata

Statesare tuples of variables’values.

Basic assumptionThe traces of a system are convergent: if two tracesreach the same state, then they evolve in the sameway

(we could use a less restrictive assumption, i.e., fusion closure)

This implies that:

the system is memoryless

a set of traces can be correctly encoded into an automaton

Tr is the set of traces under consideration.


S-System Automaton

DefinitionAn S-system automaton A(S, Tr) = (V ,∆, I, F ) is a tuple suchthat

states V = {~v = 〈v1, . . . , vn+m〉 | ∃tr ∈ Tr : ~v is in tr}transition ∆ = {(~v , ~w) | ∃tr ∈ Tr : ~v , ~w are consecutive intr}initial states I = {~v | ∃tr ∈ Tr : ~v is initial in tr} ⊆ V

final states F = {~v | ∃tr ∈ Tr : ~v is final in tr} ⊆ V


The Simpathica system

Simpathica ( Simulation of Pathways and Integrated ConcurrentAnalysis)

A tool that integrates numerical analysis, temporal logic,verification, and visualization tools and allows to:

handle biological systems described by means ofa pathwaya set of differential equations (PLAS, Octave/Mathlab, ...)a set of traces

encode the system’s behaviors into a standard automaton

easily express system’s properties to be checked

automatically verify system’s properties


The Architecture


Example: the Repressilator


The Repressilator Automaton

X1 = X4X−13 − X 0.5

1 , X4 = 0.2,

X2 = X5X−11 − X 0.578151

2 , X5 = 0.2,

X3 = X6X−12 − X 0.5

3 , X6 = 0.2.


Reducing the number of States

State explosion problemThe automata we work with may have a huge number of states

Two simple ways to reduce the automata areprojection: “ignore” some variables and consequentlyidentify states

collapsing: delete intermediate states


Projection can introduce false behaviors

Example

Projecting safely: BsimulationTwo automata are bisimilar if the states states missing in onecan be simulated by states in the other. Bisimilar automatasatisfy the same CTL-formulae.

lecture 5 the futurebortolu/files/didattica/sissacourse... · 2007-01-31 · lecture 5 the future...

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