Formal approaches to the synthesis of biological networks Nicola - - PowerPoint PPT Presentation
Formal approaches to the synthesis of biological networks Nicola Paoletti Department of Computer Science, University of Oxford Metable Workshop Computer Laboratory, University of Cambridge, 26 Mar 2015 MOTIVATION Model checking Verification
Nicola Paoletti Department of Computer Science, University of Oxford Metable Workshop
Computer Laboratory, University of Cambridge, 26 Mar 2015
MOTIVATION Model checking
Verification of given biological properties on a model M φ M | = φ?
Verification of given biological properties on a model
Model checking
M φ M | = φ?
Synthesis (CS)
Automatic derivation of a model that meets given biological properties
FROM TO
f s.t. M(f) | = φ M(·) φ
MOTIVATION
PLAN OF THE TALK
PLAN OF THE TALK
SEA URCHIN DEVELOPMENTAL GRN
and organs)
indirect contiguity) describing the evolution of embryonic geometry
(one of) THE MOST COMPLETE MODELS
Davidson et al., PNAS 109(41) 16434-6442, 2012
LIMITATIONS
Davidson et al., PNAS 109(41) 16434-6442, 2012
(discrepancies on 26/45 genes)
LIMITATIONS
Davidson et al., PNAS 109(41) 16434-6442, 2012
(discrepancies on 26/45 genes).
LIMITATIONS
SMT solving SYNTHESIS OF GRNS
How to obtain a GRN model that fully explains experimental data?
SATISFIABILITY MODULO THEORIES (SMT) SAT
SAT? + Interpretation of variables SMT (= SAT + theories)
integer/real arithmetic, …)
SAT? + Interpretation of (free) variables and functions φ = (x1 ∨ ¬x2) ∧ x3
φ = ∀x0.(x0 < x1 = ⇒ f(x0, x2) 6 x3) φ φ
expression of each gene in each domain)
Equations)
History and domain dependent
fg : Π × T × D → B (g ∈ G) r : D × D × T → B F SR T D G Q = B|G×D| Π δ : Π → B
δ(π) ⇐ ⇒ ^
i∈T,g∈G,d∈D
π[i](g, d) = fg(π, i, d)
FORMAL MODEL (GRN + DELAYS + DOMAINS)
GRN MODEL TRANSITION SYSTEM DYNAMICS
FORMAL MODEL (GRN + DELAYS + DOMAINS)
expression&of&each&gene&in&each& domain)&
Equations)&&
History&and&domain&dependent&
fg : Π × T × D → B (g ∈ G) r : D × D × T → B F SR T D G Q = B|G×D| Π δ : Π → B
δ(π) ⇐ ⇒ ^
i∈T,g∈G,d∈D
π[i](g, d) = fg(π, i, d)
GRN$MODEL$ TRANSITION$SYSTEM$DYNAMICS$
OBSERVATIONS O Wildtype expression
à predicates on paths
(e.g. gene g1 is off at time 3 in domain d1)
Perturbation experiments à
modified vector equations + predicates comparing wildtype and perturbed paths
(e.g. g1 is over-expressed in d1, time interval [5,10] and perturbation p1)
¬π[3](g1, d1) πp1[5, 10](g1, d1) > π[5, 10](g1, d1)
Synthesize functions in s.t. the dynamics of admits paths that meet all observations F N
Model encoded as constraints in the theory of bit-vectors (SMT QF_UFBV)
Input:
N = (G, D, SR, T, F)
O F PROBLEM FORMULATION
(evaluation in domain ) ¯ d (evaluation in a domain in sp. relation with ) ¯ d r (delayed permanent activation) (delayed permanent repression) (delay of steps) n FORMALIZATION OF VECTOR EQUATION LANGUAGE
E At-2 E E After-1 E
temporal
delays spatial relations domains input genes and their expression
Examples
ü Clear biological interpretation ü Incorporates uncertainty Basic interactions (BI) are templates for the synthesis of regulatory terms
f = At-[1, 3] ¬g1 f = {After-, Perm-} ? In{d1, d2} ? {g1, g2}
FUNCTION SYNTHESIS
FUNCTION SYNTHESIS
We use Uninterpreted Boolean Functions to synthesize logical combinations of regulatory inputs
(BIs or further UBFs).
temporal)
delays) spatial) relations) domains) input)genes)and)their) expression))
Examples)
! Clear)biological) interpretation) ! Incorporates)uncertainty) Basic&interactions&(BI)&are)templates)for)the)synthesis)of)regulatory)terms)
f = At-[1, 3] ¬g1 f = {After-, Perm-} ? In{d1, d2} ? {g1, g2}
Software implementation of VE language and SMT-based synthesis methods with Z3 as solving engine
User-guided refinement UNSAT core-guided relaxation
hfn1 := AT-2 bra AND AT-2 eve f1:= {AT-,AFTER-}? IN ?? bra f2:= {AT-,AFTER-}? IN ?? eve hfn1 := uf(f1,f2) f1:= AT-[0,5] bra f2:= AT-[0,5] eve hfn1 := uf(f1,f2) hfn1 := AT-1 bra AND eve Synthesized function: Original function: UF of synthesis templates:
RESULTS
FULLY EXPLAINS EXPERIMENTAL DATA
modifications
changes
f1:= {AT-,AFTER-}[0,6] IN ?? bra f2:= {AT-,AFTER-}[0,6] IN ?? eve 352800 possible functions! hfn1 := uf(f1,f2)
SUMMARY
PLAN OF THE TALK
STOCHASTIC BIOCHEMICAL REACTION NETWORKS
à stochastic dynamics à Continuous Time Markov Chains (CTMC)
STOCHASTIC BIOCHEMICAL REACTION NETWORKS AIM: Precise parameter synthesis synthesising parameters so that a given property is guaranteed to hold or the probability of satisfying is maximised/minimized
STOCHASTIC BIOCHEMICAL REACTION NETWORKS
40 1 2 3 4
Reactions and rate functions CTMC Property
(Continuous Stochastic Logic)
Parameters
Parametric CTMC (pCTMC) semantics
SATISFACTION FUNCTION
40 1 2 3 4 0.5 0.4 0.3 0.2 0.1 0.0 0.10 0.15 0.20 0.25 0.30
probability bounds
0.5 0.4 0.3 0.2 0.1 0.0 0.10 0.15 0.20 0.25 0.30
probability bounds
0.5 0.4 0.3 0.2 0.1 0.0 0.10 0.15 0.20 0.25 0.30
SYNTHESIS METHOD (SKETCH)
1) Method to compute SAFE APPROXIMATIONS to min and max probabilities over a fixed parameter region Upper and lower bounds Safe approximations
SYNTHESIS METHOD (SKETCH)
1) Method to compute SAFE APPROXIMATIONS to min and max probabilities over a fixed parameter region 2) Parameter space decomposition à improves accuracy of approximation Upper and lower bounds Safe approximations
SYNTHESIS METHOD (SKETCH)
1) Method to compute SAFE APPROXIMATIONS to min and max probabilities over a fixed parameter region 2) Parameter space decomposition à improves accuracy of approximation 3) Synthesis algorithms iterate steps 1) and 2) until required precision is reached Threshold (≥r) Max
approximation of maximum probability M
APPLICATIONS: SIR EPIDEMIC MODEL
ki
kr
Susceptible à Infected Infected à Recovered
Property: φ = (I > 0)U[100,120](I = 0)
(infection lasts at least 100 time units and ends within 120 time units)
APPLICATIONS: SIR EPIDEMIC MODEL
S + I
ki
− →I + I I
kr
− →R
Susceptible+!+Infected+ + Infected+!+Recovered+
Property:)) φ = (I > 0)U[100,120](I = 0)
(infection+lasts+at+least+100+time+units+and+ ends+within+120+time+units)+
P
ki kr
kr
ki k
0.05 0.1 0.15 0.2 0.25 0.3 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2r
Threshold synthesis:
APPLICATIONS: SIR EPIDEMIC MODEL
Max synthesis:
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4
kr ki
P
S + I
ki
− →I + I I
kr
− →R
Susceptible+!+Infected+ + Infected+!+Recovered+
Property:)) φ = (I > 0)U[100,120](I = 0)
(infection+lasts+at+least+100+time+units+and+ ends+within+120+time+units)+
APPLICATIONS: DNA WALKERS
Man-made molecular motor/robot
functions Parameters:
Dannenberg, F. et al. Natural Computing (2014)
APPLICATIONS: DNA WALKERS
Probability ≥ 80% that the walker terminates with the correct result at time 200 min AND Probability ≤ 16% that the walker terminates with a wrong result at time 200 min
Man$made(molecular(motor/robot((
functions(( Parameters:)
Property:
P>0.8[F[200,200] finish-correct]∧ P60.16[F[200,200] finish-incorrect]
APPLICATIONS: MAMMALIAN CELL CYCLE
Swat, M. et al. Bioinformatics 20(10), 1506–1511 (2004)
pRB E2F1
(A) (B)
APPLICATIONS: MAMMALIAN CELL CYCLE
Bistability property: high probability of reaching either the low (L, B<2) or the high (H, B>4) mode
P>0.4[FtL] ∧ P>0.4[FtH]
Swat,&M.&et&al.&Bioinformatics&20(10),&1506–1511&(2004)&&
pRB E2F1
(A) (B)
SUMMARY
(time-bounded) CSL
decomposition of parameter space
pRB-E2F1 circuit