Synchronous Balanced Analysis Andreea Beica, Vincent Danos cole - - PowerPoint PPT Presentation

Synchronous Balanced Analysis

Andreea Beica, Vincent Danos

École Normale Supérieure Paris GT Bioss, 13 March 2017

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Overview

What? Piecewise synchronous approximation of Chemical Reaction Networks’ (CRN) dynamics Why?

• highlight interdependence of cellular processes
• finite cellular resource allocation VS cellular processes (i.e., growth)
• rephrase mass action run of CRNs as an optimisation problem

How? Resource allocation centred Petri Nets with maximal-step execution semantics Usage?
 Approximation of real dynamics, constraint-based model similar to Flux Balance Analysis

Work in progress!

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Cellular processes rarely work in isolation: the rest of the cell cannot be ignored. Finite cellular resources: committing to one task reduces the amount available to

• thers

Define a formal notion of “growth”: use as an improved biomass function in FBA

Motivation

Andrea Y. Weiße, Diego A. Oyarzún, Vincent Danos, and Peter S. Swain Mechanistic links between cellular trade-offs, gene expression, and growth PNAS 2015 112 (9) E1038-E1047; 2015, doi:10.1073/pnas.1416533112

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Modeling CRNs

CRN = <S, ∇+,∇-, R, κ>

species

r+ 2 Rr×s

r− 2 Rr×s

reactions stoichiometry matrices

{R1, . . . , Rr}

{S1, . . . , Ss} reaction rate constants

κ : R → R>0

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PN = <P , T, W , m0 >

places transitions

8s, t : r−(s, t) = W(s, t) 8s, t : r+(s, t) = W(t, s)

r = r+ r−

W : ((S × T) ∪ (T × S)) → N initial marking

m0 : P → N

CRN mass action dynamics

ri :

s

X

j=1

r−

ijSj ki

• !

s

X

j=1

r+

ijSj

x = (xS1, ..., xSs) ∈ Ns dx dt = (r+ r)T · K · xr−

K = diag(κ1, ..., κr)

vector-matrix exponentiation

reaction: reaction network:

r−S

k

• ! r+S

system state: CRN dynamics:

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Petri Nets & Chemical Reaction Networks

3A + 2B

κ0

− → A 5B + 3C

κ1

− → 2A + 2B C

κ2

− → 2B m0 = (9, 9, 9) r− =   3 2 5 3 1   r+ =   1 2 2 2  

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Piecewise synchronous execution

Split: Burst: Collect:

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"SPLIT": resource allocation

α ∈ R|T |×|S| ∀j ∈ S, X

i∈T

αij ≤ 1

Resource allocation matrix: α ij

Interpretation: resource fraction VS reaction probability

species reaction

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"BURST": Max-parallel execution semantics of PN

“ execute greedily as many transitions as possible in one step” Definition: A max-parallel execution step in a PN at state m is a positive T-vector v s.t.:

• 1. v is compatible with m:

• 2. v is exhaustive:

0  m r−v

8j 2 T, m r−v ⇤ rj, where rj is the jth column of r−

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"BURST": Max-parallel execution semantics of PN

{t0 × 3, t2 × 9} {t0 × 2, t1, t2 × 6}

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Resource allocation & Max Parallel Execution

Define

(↵ ? m)j = min

i∈S ( ↵ji

r−

ij

· mi)

Theorem:

∀v compatible with a resource array m (and potentially max-parallel), ∃↵ resource allocation matrix s.t. v = ↵ ? m. Furthermore, if the CRN is unary, there is unicity of ↵

Our execution semantics encompasses max-parallel execution

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Growth in unary CRNs

State of the system after 1 execution with given α split: State of the system after k executions with given α split:

(I + r · α) · m

Dk

α · m, with Dα = I + r · α

Dk

α · m = λk 1 · [m1 +

X

i≥2

( λi λ1 )k · mi]

λ1 > λ2 > · · · , the eigenvalues of Dα

m = X

i

mi, with mi ∈ E(λi)

The growth rate of the system is given by λ1.

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Unary CRNs and Depletion Time

Si → . . . @k τ = k−1 log(Si(0) si ) Si → . . . @kj; j ∈ Ni, |Ni| = n τj = k−1

j

· log(αji · Si(0) si,j )

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depletion level

Unary CRNs: “iso” assumptions

Isochronous: Iso-remainder:

∀j ∈ Ni, τj = τ ∀j ∈ Ni, si,j = si

Decouple production and consumption

∆m(⌧) = r · (↵ i ✏ i) · m ∆m τ ⇡τ→0 r · [kj] · m

The usual ODE dynamics is recreated: dx

dt = (r+ r)T · K · xr−

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ˆ ↵Si(⌧, ˆ kSi) = ⇥ ↵ i − ✏ i ⇤ = ⇥ eτ·kj − 1 P

j 1 + eτ·kj

⇤

Unary CRNs: “iso” assumptions

Resource allocation matrix as a function of depletion time and reaction rate constants!

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Concrete interpretation

Approximation of system dynamics:

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τ ˆ κ α ✏ ˆ α(τ, ˆ k)

∆m(⌧) = r · (↵ i ✏ i) · m

τ → 0

dx dt = rT · K · xr−

• temporised discrete execution
• “

don’t wait for the slowest reaction”

• simulation: big step approx. of an integrator (deterministic tau-leaping)

Abstract interpretation

Synchronous Balanced Analysis

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Objective function: λ1

maximize

Dk

α · m = λk 1 · [m1 +

X

i≥2

( λi λ1 )k · mi]

αmax

ˆ α(τ, ˆ k)

Constrain reaction rates, refute models

SBA VS.FBA

• able to handle growth (no steady state assumption needed) 

• take into account real system kinetics
• characterise behaviour of a cell using only one construction: α
• replace mechanistic details of resource allocation with an abstract vector

• bjective biomass function emerges directly from method: growth rate

• maximising biggest eigenvalue of matrix: how ?

Future work

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• binary reactions: depletion time? (Michaelis-Menten

type assumption)

• explore correlations between growth rate and model

parameters

• tau-leaping, whole-cell models by Karr, etc…

Conclusion

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Piecewise synchronous approximation of the dynamics of (growth) CRNs:

parallel execution semantics of PN, based on resource allocation .

Interpretations: 1.

• approx. of real system dynamics

2. constraint based method (like FBA)

Thank you!