Analyzing the systemic function of genes and proteins

Analyzing the systemic function of genes and proteins

Rui Alves

Organization of the talk

• From networks to physiological behavior

• Network representations

• Mathematical formalisms

• Studying a mathematical model

In silico networks are limited as predictors of physiological behavior

What happens?

Probably a very sick mutant?

How to predict behavior from network?

• Build mathematical models!!!!




• Mathematical formalisms


Network representation is fundamental for clarity of analysis

A B

What does this mean?

Possibilities:

AB

Function

BA

Function

AB

Function

A B

Function

BA

Function

Defining network conventions

A B

C

Full arrow represents a flux between A and B

Dashed arrow represents modulation of a flux

+

Dashed arrow with a plus sign represents positive modulation of a flux

-

Dashed arrow with a minus sign represents negative modulation of a flux




• Mathematical formalism


Representing the time behavior of your system

/dA dt

/ ,dA

dA dt A f A Cdt

/

dAdA dt

dt

A B

C

+

/dA

dA dt Adt

What is the form of the function?

A B

C

+

A or C

Flux1 2k A k CLinear 1 2

1 2 3 4

k A k C

K K A K C K AC

Saturating

4 41 2

44 41 2 3 4

k A k C

K K A K C K AC

Sigmoid

What if the form of the function is unknown?

A B

C

+

int intintint int

2 2

2int int

int

, ,, ,

, ,

operating operatingoperatingpo popo operating operating

po po

operating operatingpo pooperating

po

df A C df A CdAf A C f A C A A C C

dt dA dC

d f A C d f A CA A C C

dAdC d C

2

int

int

2 2

2int

int

,...

operatingpooperating

po


po

C C

d f A CA A

d A

Taylor Theorem:

f(A,C) can be written as a polynomial function of A and C using the function’s

mathematical derivatives with respect to the variables (A,C)

Are all terms needed?

A B

C

+

int

intint

intint

, ,

,

,

operatingpo


po


po

dAf A C f A C

dt

df A CA A

dA

df A CC C

dC

f(A,C) can be approximated by considering only a few of its mathematical derivatives

with respect to the variables (A,C)

Linear approximation

A B

C

+

1 20, ,

dAf A C f A C k A k C

dt

Taylor Theorem:

f(A,C) is approximated with a linear function by its first order derivatives with respect to

the variables (A,C)

What if system is non-linear?

• Use a first order approximation in a non-linear space.

Logarithmic space is non-linear

A B

C

+

1 2, g gdAf A C A C

dt

g<0 inhibits flux

g=0 no influence on flux

g>0 activates flux

Use Taylor theorem in Log space

Why log space?

• Intuitive parameters

• Simple, yet non-linear

• Linearizes exponential space

–Many biological processes are close to exponential → Linearizes mathematics

Why is formalism important?

• Reproduction of observed behavior

• Tayloring of numerical methods to specific forms of mathematical equations






A model of a biosynthetic pathway

10 13 111 1 0 3 1 1/ g g hdX dt X X X

11 222 1 1 2 2/ h hdX dt X X

X0 X1

_

+

X2 X3

X4

22 33 343 2 2 3 3 4/ h h hdX dt X X X

Constant

Protein using X3

What can you learn?

• Steady state response

• Long term or homeostatic systemic behavior of the network

• Transient response

• Transient of adaptive systemic behavior of the network

What else can you learn?

• Sensitivity of the system to perturbations in parameters or conditions in the medium

• Stability of the homeostatic behavior of the system

• Understand design principles in the network as a consequence of evolution

Steady state response analysis

10 13 111 1 0 3 1 1/ 0g g hdX dt X X X

11 222 1 1 2 2/ 0h hdX dt X X

22 33 343 2 2 3 3 4/ 0h h hdX dt X X X

How is homeostasis of the flux affected by changes in X0?

0 3 0 10 33 13( , ) ( , ) /L V X L X X g h g

Log[X0]

Log[V]

Low g10

Medium g10

Large g10

Increases in X0 always increase the homeostatic values of the flux through the pathway

How is flux affected by changes in demand for X3?

4 13 34 13 33( , ) / 0L V X g h g h

Log[X4]

Log[V]Large g13

Medium g13

Low g13

How is homeostasis affected by changes in demand for X3?

3 4 4 13 34 13 33( , ) ( , ) / / 0L X X L V X g h g h

Log[X4]

Log[X3]

Low g13

Medium g13

Large g13

What to look for in the analysis?

• Steady state response

•Long term or homeostatic systemic behavior of the network

• Transient response

•Transient of adaptive systemic behavior of the network

Transient response analysis

10 13 111 1 0 3 1 1/ g g hdX dt X X X

11 222 1 1 2 2/ h hdX dt X X

22 33 343 2 2 3 3 4/ h h hdX dt X X X

Solve numerically

Specific adaptive response10 13 11

1 1 0 3 1 1/ g g hdX dt X X X 11 22

2 1 1 2 2/ h hdX dt X X 22 33 34

3 2 2 3 3 4/ h h hdX dt X X X

Get parameter valuesGet concentration

valuesSubstitution

Solve numerically

Time

[X3]

Change in X4

General adaptive response10 13 11

1 1 0 3 1 1/ g g hdX dt X X X 11 22

2 1 1 2 2/ h hdX dt X X 22 33 34

3 2 2 3 3 4/ h h hdX dt X X X Normalize

Solve numerically with comprehensive scan of parameter values

Time

[X3]

Increase in X4

Low g13

Increasing g13

Threshold g13

High g13

Unstable system, uncapable of homeostasis if feedback is strong!!

Sensitivity analysis

• Sensitivity of the system to changes in environment–Increase in demand for product causes increase in flux through pathway

–Increase in strength of feedback increases response of flux to demand

–Increase in strength of feedback decreases homeostasis margin of the system

Stability analysis

• Stability of the homeostatic behavior

–Increase in strength of feedback decreases homeostasis margin of the system

How to do it

• Download programs/algorithms and do it– PLAS, GEPASI, COPASI SBML suites,

MatLab, Mathematica, etc.

• Use an on-line server to build the model and do the simulation– V-Cell, Basis

Design principles

•Why is a given pathway design prefered over another?

•Overall feedback in biosynthetic pathways

•Why are there alternative designs of the same pathway?

•Dual modes of gene control

Why regulation by overall feedback?

X0 X1

_

+

X2 X3

X4

X0 X1

_

+

X2 X3

X4

__

Overall feedback

Cascade feedback

Overall feedback improves functionality of the system

TimeSpurious stimulation

[C]Overall

Cascade

Proper stimulus

Overall

Cascade

[C]

StimulusOverall

Cascade

Dual Modes of gene control

Demand theory of gene control

Wall et al, 2004, Nature Genetics Reviews

• High demand for gene expression→ Positive Regulation

• Low demand for gene expression → Negative mode of regulation

How to do it

• Download programs/algorithms and do it– BST Lab, Mathematica, Maple

Summary





Papers to present

• Vasquez et al, Nature

• Alves et al. Proteins

Computational tools in Molecular Biology

• Predictions & Analysis– Identification of components– Organization of components– Conectivity of components– Behavior of systems– Evolution & Design

• Prioritizing wet lab experiments– Most likely elements to test– Most likely processes to test

The Taylor theorem

C

f(C)

0 order

f(C)

1st order

2nd order

ith order

ith + jth order

Are all terms needed?

A B

C

+

int

intint

intint

, ,

,

,

operatingpo


po


po

dAf A C f A C

dt

df A CA A

dA

df A CC C

dC

f(A,C) can be approximated by considering only a few of its mathematical derivatives

with respect to the variables (A,C)

Linear approximation

A B

C

+

1 20, ,

dAf A C f A C k A k C

dt

Taylor Theorem:

f(A,C) is approximated with a linear function by its first order derivatives with respect to

the variables (A,C)

What if flux is non linear?

A B

C

+

Use Taylor theorem in Non-Linear space!

Use Taylor theorem with large number of terms

or

How does the transformation between spaces work?

0 1

0

X

Y

2 2 21/ 4X Y

X

Y

X

Y

How does the Taylor approximation work in another space?

Variables:

A, B, C, …

f(A,B,…)

Variables:

A, B, C, …

f(A,B,…)

~f(A,B,…)

Taylor theorem

Transform to new space

Return to original space

~f(A,B,…)

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Analyzing the systemic function of genes and proteins