mesoscopic and stochastic phenomena and the lac operon

Towards Systems Biology, October 2007, GrenobleAdam Halasz

Ádám Halász

joint work with

Agung Julius, Vijay Kumar, George Pappas

Mesoscopic and Stochastic Phenomena and the Lac operon


Outline

• Lactose induction in E. coli, an example of bistability

• Stochastic phenomena in reaction networks

• Mesoscopic effects in the lac operon

• Outlook


Lac system: Biological phenomenology

lac Z lac Y lac A

lac I

repressor

mRNAmRNA

permease

externallactose

internallactoseallolactose

galactosidase


Lac system: ODE model

Externallactose

LactoseL

AllolactoseA

β-galactosidaseB

PermeaseP

mRNAM

• Network of 5 substances

• Example of positive feedback in a genetic network discovered in the 50’s

• Described by differential equations which are built from chemical rate laws

• Some time delays and time scale separations ignored and/or idealized


Lac system: ODE model

• Network of 5 substances

• Example of positive feedback in a genetic network discovered in the 50’s

• Described by differential equations which are built from chemical rate laws

• Some time delays and time scale separations ignored and/or idealized

External TMGTe

TMGT

β-galactosidaseB

PermeaseP

mRNAM

PMdt

dP

TTK

TP

TK

TP

dt

dT

BMdt

dB

MTKK

TK

dt

dM

PP

TL

LeT

eL

BB

MM

e

)(

)(

)(

)(1

021

21


Lac system: ODE model• Because of the positive feedback, the

system has an S-shaped steady state structure

• That is to say, for some values of the external inducer concentration (Te), there are two possible stable steady states

Pin

Pout

Texternal

Bequilibrium

Te

B P


Lac system: Bistability

• Switching and memory– Need to clear T2 in order to switch up

Te

B

T1 T2

B

Te

t

t


Lac system: Bistability• Switching property is robust

– Model parameters perturbed by 5%


Lac system – first lesson

• Network of <5 species involved in reactions• Reactions decomposed* into mass action laws• Can be implemented using simple stochastic

transition rules eg: “upon colliding with a B, A becomes C with probability x”

• Recipe for synthesis of switch with hysteresis• Compose motifs to build logical functions,…

– Design the network and transition rules– Map the rules to individual stochastic programs


Lac system: ODE is not enough


Stochastic versus deterministic• Substances are represented by finite

numbers of molecules• Rate laws reflect the probability of

individual molecular transitions• The abstraction/limit process is not

always trivial

BA k

][][

Akdt

Ad

t1 N1 t2 N2=N1-2

tktttPtBA 0

),(

tktNtttN ANtA A )(),(

,0

kep )(Next transition time distribution::


Chemical reactions are random events

A

B

A + B AB A + B AB

A

B


Stochastic reaction kinetics

• Quantities are measured as #molecules instead of concentration.

• Reaction rates are seen as rates of Poisson processes.

• A + B AB

k

Rate of Poisson process


Stochastic reaction kinetics

reaction

time

time

A

AB

reaction reaction


Multiple reactions

• Multiple reactions are seen as concurrent Poisson processes.

• Gillespie simulation algorithm: determine which reaction happens first.

A + B ABk1

k2

Rate 1 Rate 2


Multiple reactions

reaction 1

time

time

A

AB

reaction 2 reaction 1


Gillespie simulations• When the number of molecules per cell

is small*, the respective substance has to be treated as an integer variable

• The probabilistic transition rules can be implemented in standard ways

• Gillespie method: instead of calculating time derivatives, we calculate the time of the next transition

• Many other sophisticated methods exist. As an empirical rule, the higher the number of molecules, the closer the simulation is to the continuous version


Gillespie automata

• The state of the system is given by the number of copies of each molecular species

• Transitions consist of copy number changes corresponding to elementary reactions

• The distribution of the next transition time is Poisson, e-kt where k is the propensity

• A Gillespie automaton is a mathematical concept [a continuous time Markov chain]

• Plays the same role differential equations have in the continuum description


Lac system: spontaneous transitions both ways

External TMG concentration

mR

NA

co

ncen

trat

ion

Time (min)

0 500 1000 15000

5

10

15

20

25

30

35#

mR

NA

mol

ecul

es

Increase E


Mixed Gillespie/ODE

1000 cells

Aggregate simulations


• We create a simplified model, a continuous time Markov chain with two discrete states, high state and low state

• The transition rates depend on the external concentration of TMG

Two state Markov chain model

Lo Hi

(Julius, Halasz, Kumar, Pappas, CDC06, ACC07)


Transition rates

Identified transition rates(Monte Carlo)


Two state Markov chain model

0 500 1000 15000

5

10

15

20

25

Average of a colony with 100 cells

Time (min)

# m

RN

A m

olec

ules

E[M ]t


Lac system – second lesson

• Bad news: underlying stochasticity can drastically modify the ODE prediction– The price paid for an ‘uncontrolled’ approximation

• Good news: the ODE abstraction can also be performed rigorously– Gillespie automaton is an exact abstraction– For Gillespie ODE, all ‘molecule’ numbers must be large

• Stochastic effects retained at the macro-discrete level– Effects are reproducible and quantifiable– Further abstractions of stochastic effects are possible

• Lac example: can quantify the spontaneous transitions:– Choose an implementation where they are kept at a low rate– Implement control strategies that use the two-state model


Beyond Gillespie

• Gillespie method is ‘exact’ – produces exact realisations of the stochastic process

• Main problem is computational cost– Larger molecule numbers– Rare transitions

• Several approaches to circumvent ‘exact’ simulations


Towards the continuum limit

– leaping: lump together several transitions update molecule numbers at fixed times:

r1

time

time

AB

r2 r1 r1r1

r2 r2

A


Towards the continuum limit

• Error introduced by – leaping is due to variation of the propensities over the time interval(may lead to negative particle numbers!)

• Acceptable if the expected relative change of each particle number over Δ is small (e.g. if the number of particles is large)

• If the number of transitions per interval is also large, the variation can be described as a continuous random number stochastic differential equations

• Finally, is the variance of the change per interval can also be neglected, the simulation is equivalent to an Euler scheme for an ODE.


Other limiting cases

• If the number of all possible configurations is relatively small, probabilities for each state can be calculated directly, by calculating all possible transition rates, (finite state projection) or using the master equation (Hespanha, Khammash,..)

• In some situations (eg. signaling cascades) there is a combinatorial explosion of species, where agent-based simulations are useful (Los Alamos group, Kholodenko)


Transitions in the lac system

• We used tau-leaping for our simulations

• The high state can be simulated using SDEs or ‘semiclassical’ methods

• The lower state can be studied using finite state projection

Time (min)

# m

RN

A m

olec

ules

0 500 1000 15000

5

10

15

20

25

30

35


Summary

• Mesoscopic effects in biological reaction networks are due to small numbers of molecules in individual cells

• They may affect the system dramatically, somewhat, or not at all

• These effects can be described mathematically and incorporated in our modeling efforts

• Several sophisticated methods exist; it is important to use an approximation that is appropriate, both in terms of correctness and in terms of efficiency


Thanks:

Agung Julius, , George Pappas, Vijay Kumar, Harvey Rubin

DARPA, NIH, NSF, Penn Genomics Institute

mesoscopic and stochastic phenomena and the lac operon

Documents

uptebt1t2btettlac system

lac operonoutlooklac

lessonnetwork of lac

transition time distribution

time derivatives

time scale separations

andor idealizedlac system

reaction rates