Mathematical Modeling of

Signal Transduction Pathways

1

1

. .( )

( )

k T

m T

k E Y YpdYp dt

K Y Yp

Biplab Bose

IIT Guwahati

Cellular Communication

Ligand

Receptor

Rel

ay

Output

Message

FunctionE

ncod

ing/

Dec

odin

g

Image: BioCarta

Encoding-decoding in Dynamics

Nat Rev Mol Cell Biol. 2011, 12(2):104-17.Cell. 1995, 80(2):179-85.

Why Model?

• To understand empirical observations

• To generate new hypothesis

Linear Network Network with Negative Feedback

Signaling beyondsaturation

PLoS Comput Biol 4(10): e1000197

Weber’s Law

Weber’s Law in Signaling

min tanbackground

sCons t

S

Circuit that senses only fold change

Mol Cell. 2009, 36(5):724-7

Deterministic Modeling

• Assumes that the system is large

• Uses Law of Mass Action

• Homogenous system: Ordinary Differential

Equation (ODE)

• Non-homogenous system: Partial Differential

Equation (PDE)

1 2

[ ][ ][ ] [ ]

d Ck A B k C

dt

production degradation

3 4

3 4

[ ]([ ] [ ]) [ ][ ][ ]

([ ] [ ]) [ ]T

m T m

k C D pD k E pDd pD

dt k D pD k pD

Deterministic Modeling

System of ODEs

[A], [B] and [E] considered constant

Deterministic Modeling

Solve the ODEs

Analytical solution

Numerical solution

[C]

[pD]

1 2

[ ][ ][ ] [ ]

d Ck A B k C

dt

3 4

3 4

[ ]([ ] [ ]) [ ][ ][ ]

([ ] [ ]) [ ]T

m T m

k C D pD k E pDd pD

dt k D pD k pD

Modeling Strategy

Model

Data

Estimate parameters

Simulate to predict

Images: Mol Cell. 2012, 46(6):820-32.

X

Yp

Sustained signaling: both X and Yp reach steady state

Simple but Complex

The system has memoryCan lead to two population of cells

Simple but Complex

IRS1

PI3K

Akt

mTOR

Insulin/IGF-1

The mTor Story

Nat Rev Drug Discov. 2007,6(11):871-80

Database Web Address

KEGG http://www.genome.jp/kegg

REACTOME http://www.reactome.org/ReactomeGWT/entrypoint.html

PATHWAY INTERACTION DATABASE (PID)

http://pid.nci.nih.gov

PANTHER http://www.pantherdb.org/pathway

WikiPathways http://www.wikipathways.org

SMPDB http://www.smpdb.ca

Pathway Database

The parameters

http://bionumbers.hms.harvard.edu/

Database Web Address

BioModels http://www.ebi.ac.uk/biomodels-main

CellML http://www.cellml.org/models

JWS Online http://jjj.biochem.sun.ac.za/index.html

Model Database

Tools forDynamic Simulation

• JSim• COPASI• GEPASI• CellDesigner

• MATLAB• Mathematica

Extended list: Biochimie. 2006, 88(3-4):277-83.

Stochasticity in Chemical Reaction

Conventional reactions involve large number of molecules

A + B C

[ ].[ ].[ ]

d Ck A B

dt

Follows Law of Mass Action

When number of molecules is low

Can not apply Law of Mass Action

Uncertainty in reactions

Some misconceptions about random/stochastic process:

Any thing can happen.

Things are mixed-up

Does not have cause-and-effect relation.

We can not make predictions.

What is Random?

What is Random?

Random walk in Brownian motion

1. Water molecules are in motion.

2. Hit each other and the pollen.

3. Classical mechanics can be used to

understand (approximately)

trajectories due to collisions.

Facts:

Problem:

We do not (or can not have) have exact information of the system (ie. position and momentum of each particle)

Consequence:1. Can not predict exact trajectory. 2. Get surprised by the movement of the pollen3. Call this random or stochastic process

Image source: wikipedia

Beyond uncertainties

1. We can calculate average

behaviour

2. We can calculate

probability of an outcome

3. We can calculate

distribution of outcomes F

requ

ency

Displacement

Can not calculate exact displacement of a pollen in a particular duration

But Can calculate the PROBABILITY of a particular amount of displacement.

What is Random?

Protein expression similar to coin toss:You may get Head or Tail in a tossAt one moment cell may make one copy protein or not

1 2 3 4 5

1 2 3 4 5

Cell 1

Cell 2

Y Y Y

Y YY Y

N N

N

1 2 3 4 5

time

time

time

Pro

tein

num

ber

1

2

3

4

5

Protein Expression Like Coin Toss

Stochasticity in Gene Expression

Linear Circuit

Positive feedback

Transcriptional Circuit Affects Expression Heterogeneity

PLoS ONE. 2015 10(2): e0116748 

Modeling Stochastic Systems

Kinetic Monte Carlo

Gillespie algorithm

MATLAB

Dizzy

StochSim

STOCKS

Constrains of Dynamic Modeling

Difficult to model very large system

Difficulty in parameter estimation:

How to design experiment?How to estimate parameters?

Problem in connecting different scales

biplabbose@iitg.ernet.in

@bose_biplab

http://flowpy.wikidot.com/

FlowPy

How a cell handles information