Computational Science and Engineering!

Stochastic Modeling in Systems Biology

Yang Cao

Department of Computer Science

Computational Science and Engineering!Lambda-phage affected E. Coli

Computational Science and Engineering!Highlight the lambda phage regulation

cI

PL

PR N cro

PRM

cI Cro

If cI wins, PR and PL are repressed and the cell enters lysogeny

If Cro wins, PRM is repressed and the cells enters the lytic cycle

Computational Science and Engineering!

A close up on the right promoter- operator region

PRM

Computational Science and Engineering!cI must bind to OR1 to repress rightwards transcription

cI represses PR – shuts off cro

cI activates PRM – expression of cI

PRM

Computational Science and Engineering!Cro must bind to OR3 to repress expression of repressor by PRM

Cro represses PRM – shuts off cI expression

PRM PRE

Computational Science and Engineering!Lambda-phage affected E. Coli

Stochastic effects play an important role

in lytic/lysogenic decision network

Arkin et al. 1997, 1998

Lysis

Lysogeny

Computational Science and Engineering!Chemical Reacting System

Computational Science and Engineering!A General Question

Computational Science and Engineering!Molecular Dynamics

Computational Science and Engineering!Some Facts about Cell (got from Mark Paul)

Computational Science and Engineering!

Computational Science and Engineering!

Computational Science and Engineering!Characteristics of the System

Computational Science and Engineering!One Reasoning (but not a proof)

Computational Science and Engineering!Propensity Function vs the Reaction Rate

•  Difference

–  Propensity function describes the probability while reaction rate describes the changing rate.

–  Propensity functions are defined based on population of species while the reaction rates are defined based on the concentration of species

•  Connection

–  For simple system, they have similar format. For reaction like

•  Reaction rate:

•  Propensity function and we have

–  For reaction like

•  Reaction rate •  : •  Propensity function and we have

]][[ BAk

BA→][AkAcx kc =

CBA →+

BAxcx Vkc /=

Computational Science and Engineering!Chemical Master Equation

Computational Science and Engineering!SSA

Computational Science and Engineering!Direct Method (DM)

(Inverse Generation Method)

Computational Science and Engineering!First Reaction Method (FRM)

•  A different but equivalent simulation method for SSA •  Generate a firing time for each reaction channel

•  Find the minimum of all the firing time and throw off all the others

•  In theory this method is equivalent to the DM •  The biggest concern is the computational cost

,1ln)(

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

jjj rxa

τ

=µ the index satisfying

, min jjττ µ =

Computational Science and Engineering!Next Reaction Method (NRM)

•  Derived by Gibson and Bruck 2000 •  Keep the randomly generated firing time if the

corresponding propensity function is not changed ( Use a dependent graph (DG) to achieve this goal )

•  Use absolute time instead of relative time

•  Reuse the unapplied uniform random number

•  Use a priority tree (heap array) to conduct the search •  Only need to generate one uniform random number each

step •  The computational cost to maintain the data structure can

be hug!

Computational Science and Engineering!Optimized Direct Method (ODM)

•  Analyze the time profile of DM and NRM. Conclude that the data structure maintaining cost is very high for most problems.

•  Sort the reaction channel so that more frequent firing reaction channels have smaller index. (Run a few pre-simulation to collect info for the problem. )

•  If necessary, use the dependent graph (DG) to avoid recalculating propensity functions for reaction channels that are not affected by the last reaction.

•  Efficient for multiscale problem •  Among all test problems we tried, ODM is faster than NRM.

Computational Science and Engineering!Sorted Direct Method (SDM)

•  The pro-simulation procedure is troublesome for an automatic code.

•  The index should be changed dynamically during the simulation •  Bubble sorting technique

If a reaction just fired, move its index one step up. After a while, the reaction index will be automatically well-sorted.

•  Less (but almost the same) efficient than ODM but much easier to code and maintain.

Computational Science and Engineering!

Computational Science and Engineering!A Model for Prokaryotic Gene Expression

1.  Transcription Initiation (the binding and initiation)

2.  Elongation (RBS is available before elongation terminates

3.  Translation Initiation

4.  Elongation

RNAPP RNAPP +→•

RNAPPRNAPP •→+

RNAPTr RNAPP →•

ElRNAPPRBSTrRNAP ++→

RibRBSRBS Ribosome →+

RBS RibosomeRibRBS +→

RBS ElRibRibRBS +→

decayRBS→

Protein ElRib→decayProtein→

1-181 M10 −= sk

12 10 −= sk

13 1 −= sk

14 1 −= sk

1-185 M10 −= sk

16 25.2 −= sk

17 5.0 −= sk

18 3.0 −= sk

19 015.0 −= sk

1510 1042.6 −−×= sk

Computational Science and Engineering!Simulation Results

Kierzek, A. M. et al. J. Biol. Chem. 2001;276:8165-8172

Computational Science and Engineering!� Model and Error Measurement

•  Reactions:

•  Propensity functions:

•  Bistable distribution

XB

XXB

c

c

c

c

↔

↔+

3

4

1

2

2

1 32

.)(,)(

),2)(1(6

)(

),1(2

)(

44

233

22

11

1

xcxaNcxa

xxxcxa

xxNcxa

=

=

−−=

−=

Histogram plot of the state in model

gloSchl

gloSchl

Computational Science and Engineering!Single Simulation Results

Computational Science and Engineering!Stochastic Modeling

Lotka reactions:

ZYYYXXXA

c

c

c

⎯→⎯

⎯→⎯+

⎯→⎯+

3

2

1

22

Lead to ODEs

⎩⎨⎧

+−=

−=

yxccyxycAcx)()(

23

21

The stochastic simulation generates interesting trajectories.

10,01.0,10

3

2

1

=

=

=

ccAc

Computational Science and Engineering!Brusselator

DYYYXCYXB

XA

c

c

c

c

⎯→⎯

⎯→⎯+

+⎯→⎯+

⎯→⎯

4

3

2

1

32

⎪⎩

⎪⎨⎧

−=

−+−=

yxBycy

xcyxBycAcxc

c

222

42

221

3

3

.5,00005.0

,50,5000

4

3

2

1

=

=

=

=

ccBcAc

Lead to ODEs

.5,0001.0

,50,5000

4

3

2

1

=

=

=

=

ccBcAc

Bifurcation happens around the condition:

( )3

424

212 2

3

2

cc

cAc

cBc +=

J. Tyson’s 1973, 1974 paper

Computational Science and Engineering!Break the Assumption