http://creativecommons.org/licenses/by-sa/2.0/

Prof:Rui Alvesralves@cmb.udl.es

973702406Dept Ciencies Mediques Basiques,

1st Floor, Room 1.08

From networks to physiological behavior

Graphical network representations

Types of problems

Typical bottlenecks and assumptions in model building

What happens?

Probably a very sick mutant?

Maybe resolving ambiguity in representation is enough to predict

behavior?

X0 X1 X2 X3

X0

X1

X2

X3

t0 t1 t2 t3

X0 X1 X2 X3

Build mathematical models!!!!

From networks to physiological behavior

Graphical network representations

Types of problems

Typical bottlenecks and assumptions in model building

A B

What does this mean?

Possibilities:

AB

Function

BA

Function

AB

Function

A B

Function

BA

Function

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

A and B – Dependent Variables

(Change over time)

C – Independent variable

(constant value)

A B

C

Stoichiometric information needs to be included

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

23 D+

Reversible Reaction

B

C

Stoichiometric information needs to be included

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

2 A3 D

C

Having too many names or names that are closely related my complicate interpretation and set up

of the model.

Therefore, using a structured nomenclature is important for book keeping

Let us call Xi to variable i

AB

DX3

X1X2

X4

X2

X3

+2 X13 X4

C

AB

DX3

X1X2

X4

X2X0

Production Reaction

Sink Reaction

X2X0

Organel

Cell

Compartmental models are important, both because compartments exist in the cell and because even in the absence of compartments reaction media are

not always homogeneous

Whatever representation is used be sure to be consistent and to know exactly what the different elements of a representation mean.

/dA dt

/ ,dA

dA dt A f A Cdt

/

dAdA dt

dt/dA

dA dt Adt

A B

C

+

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

From networks to physiological behavior

Network representations

Types of problems

Typical bottlenecks and assumptions in model building

That depends on the question!!!!

It also depends upon the system for which you ask the question!!!!

The big one: How does a cell work???

What answers are being given?

Genome sequenced and annotated

Map onto cellular circuits chart

Create stoichiometric

model.

dXS v

dt0 .S v

stoichiometric matrix

rate vector

Usually solved for steady state

1. Assume that cells are growing at steady state with some optimal conversion of input material (flux b1) into biomass (A,B,C)

2. Assume linear kinetics for each rate equations

3. Use (linear) optimization methods to find a solution for the distribution of fluxes that allows the cell to fulfill 1.

Accurately predicting a decent fraction of knock out mutants that are lethal in S.cerevisiae and H. pylori.

Proc Natl Acad Sci U S A. 100: 13134-13139; J Bacteriol. 184: 4582-4593.

Fail to predict all mutants Does not account for transient behavior Does not account for dynamic regulationWhole cell modeling is far from being able

to answer the big question; not enough info is available to build the models.

(?)

Well, let us be modest: How does a simple cell work???

What is a simple cell? A cell that is much simpler than what we normally think

of as a cell Red Blood cell; lambda phage Mathematical models using dynamic equations have been created

to study these types of cells. (e.g. Ni & Savageau or Arkin ) A regular cell that we represent in a simplified maner

E-cell project represents the E. coli cell using linear kinetics.

. . ( , )dX

S v S f p Xdt

Savageau & Ni, 1992 JBC, JTB

Model was used to assess how complete our understading of red blood cell metabolism is.

How was this done? Using the notion that model robustness can be

used to identify ill defined parts of the model Using the notion that biological systems

should have stable steady states

Robustness is the notion that the dynamic behavior of a system is fairly insensitive to spurious fluctuations in parameter values

Parameter (T, kinetic parameters)

Ste

ad

y st

ate

va

lue

Because if biological systems were not robust, we would not be alive, given that fluctuations happen all the time.

Parameter (T, kinetic parameters)

Ste

ad

y st

ate

va

lue

Stability of a steady state is the notion that after spurious fluctuations in parameter values, the system will return to the original steady state it was in

t

X

Again, because if biological systems were not at stable steady states, we would not be alive, given that fluctuations happen all the time.

Found that the steady state was unstable Identified regulatory interactions that

stabilized the steady state Latter confirmed experimentally

Identified parts of the model that have high sensitivy Incomplete understanding of the system

Well, yes there are. There is a fair amount of modularity in

cells Organeles, Pathways, Circuits, etc.

Therefore, if one is interested in specific parts of cellular function and response, one can isolate the modules responsible for that function or response How does the specific part of a cell

responsible for a given function works???

• How does the specific part of a cell responsible for a given function works???

How does it work qualitatively Network reconstruction (RA)

P1 P2

P…

Pn

M1

M2M…

Mn

• How does the specific part of a cell responsible for a given function works???

How does it work qualitatively Network reconstructionP1

P2

P… Pn

M1M2

M…

Mn

FeSC biogenesis is a pathway that is conserved over evolution

Proteins involved in the pathway are identified

How these proteins act together to form a pathway is unknown; the reaction topology and the regulatory topology is unknown

How do these proteins work together?

Create all possible topologies

Scan all possible behaviors using simulation

Compare qualitative dynamic behavior of the different topologies to experimental results

Eliminate topological alternatives that do not reproduce experimental results

•Alves et. al. 2004 Proteins 57:481

•Vilella et. al. 2004 Comp. Func. Genomics 5:328

•Alves et. al. 2004 Proteins 56:354

•Alves & Sorribas 2007 BMC Systems Biology 1:10

Prediction Verified?

Grx5 modulates Nfs1 and Scaffold activity/Interactions

Detected interaction with scaffolds

Arh1-Yah1 act on S or ST Yes [PNAS 97:1050; JBC 276:1503]

Arh1-Yah1 interaction same as in mammals

No reported experiment

Yfh1 acts on S, T, or ST Yes [Science 305:242; EMBO Rep 4:906; JBC 281:12227; FEBS Lett

557:215]

Yfh1 storage of Fe not important for its role in biogenesis

Yes [EMBO Rep 5:1096]

Nfs1 acts in S, not necessarily in R No reported experiment

Chaperones act on Folding, Stability

Yes for Folding [JBC 281:7801]

•Alves et al. 2008 Curr. Bioinformatics in press

• How does the specific part of a cell responsible for a given function works???

How does it work quantitatively Parameter estimation when network is

known (PM)P1

P2

P… Pn

M1M2

M…

Mn

• If you know the topology and/or mechanism, then one can ask how does a system act under specific circumstances

• To answer such a question we often need numerical values for the parameters of the system so that simulations can be ran

• Numerical values for parameters can be estimated from experimental data

• Based on gene expression data, what are the parameter values that create a best fit of the model to the observed experimental results?

Collect experimental data

Create a mathematical model

Use optimization/fitting methods to estimate the parameters of the model in such a way that a minimum discrepancy exists betrween model predictions and observed data.

Hell, No!!!!

Modularity begs the question: Are there design principles that

explain why cell use specific modules for specific functions? [AS; RA;AS]

X0 X1

_

+

X2 X3

X4

X0 X1

_

+

X2 X3

X4

__

Overall feedback

Cascade feedback

Create mathematical models for the alternative networks

Compare the behavior of the models with respect to relevant functional criteria

Decide according to those criteria which model performs best

TimeSpurious stimulation

[C]Overall

Cascade

Proper stimulus

Overall

Cascade

[C]

StimulusOverall

Cascade

Alves & Savageau, 2000, Biophys. J.

From networks to physiological behavior

Graphical network representations

Types of problems

Typical bottlenecks and assumptions in model building

• Is your system the whole cell? If so, how detailed do you want to make your

model to be?

• Is your system the whole cell? If so, how detailed do you want to make your

model to be?• If your system is not the whole cell, is it a

pathway or circuit? How do you define pathway? What will you include in your model?

Include cofactors, elementary steps?

Include all reactions? Not all are present in a

given organism

• No magic bullet exists to define your system. Read the literature, learn about your system,

guesstimate the important inputs and bound your system as a module: Simplify as much as you can but not more than that

• Should we include all details known about the system?

• What can we simplify?

• Again, no general answer for this. Read the literature, learn about your system,

guesstimate the important inputs and bound your system as a module

What is the form of f(p,X)?

. . ( , )dX

S v S f p Xdt

Individual steps of all processes are mass action

The kinetics of a process may be complicated

( , ) 1 jn

m i i ji j

f p X k X

X1 X2

P1

X1

P1

X1P1 X2P1

X2

P1

We may end up with a model that is larger than it has to be:

2 Variables 4 Variables

Use rational enzyme kinetics to reduce model dimension: e.g. HMM kinetics

( , )mV X

f p XK X

X1 X2

P1

X1

P1

X1P1 X2P1

X2

P1

Allow for dimensional reduction of models while often still being accurate

Form is mechanism dependent Michaelis-Menten, Hill, Theorel-chance, etc.

Assumes that E<<<S and/or very different time scales for the individual processes E.g. in signal transduction, In some cases time scale simplification is incorrect

E S

Usually we do not know the individual mechanistic steps of processes

Therefore, using rational enzyme kinetics is not justified

However, one can use approximate formalisms Power Law, Saturating cooperative formalism,

etc…

Form is always the same (if Taylor based) Automated equation building from graphical

representation

Parameters are fairly easy to estimate

One needs to choose the appropriate formalism for the specific situation E.g. if a process saturates, one may use a piece-

wise power law or a SC formalism equation

X1 X2

P1

X1

v

Power Law

Piece Wise Power Law

Saturating Cooperative formalism

Lineal

Piece wise

If Taylor based, they are absolutely accurate only at the operating point of the approximation However, they may have a range of sufficient

accuracy of several orders of magnitude about the operating point

How do we analyze this? If closed form solutions are available, analysis may be

made independent of parameter values

Closed form solutions are almost never available!!!! Lineal approximations allways have close form solutions Power law, other transformations may also have closed

form steady state solutions

. . ( , )dX

S v S f p Xdt

If parameter values are available, then solutions can be numerically calculated (PM)

Numerical solutions allows us to predict the behavior of a specific system

. . ( , )dX

S v S f p Xdt

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

X0 X1

_

+

X2 X3

X4

Constant

Protein using X3

• Steady state response

• Long term or homeostatic systemic behavior of the network

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

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

• Stability of the homeostatic behavior of the system

• For both, you only need to know how to do derivatives!!!!

• Transient response

•Transient of adaptive systemic behavior of the network

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

In and of itself a model is a model. It needs to be contrasted to reality

If when contrasted to reality, model predictions are verified, the model is validated; otherwise it is back to the drawing board

Models are never valid under all conditions

• All molecular species are present in discrete ammounts within a cell

• If one assumes that sufficiently large ammounts are present, it is OK to treat species as concentrations/densities, thus simplifying calculations => Deterministic ODE models

• No answer is always right for this question

• However, if small number of particles is involved in the process, assumption breaks down

• How to solve this problem? ‾ Either use statistical master equation or

stochastic differential equations

• If one assumes that all cellular compartments are well mixed in a time scale faster than the processes of interest, it is OK to use ODE models, either deterministic or stochastic

• There are all sorts of compartments and gradients within a cell

• Often, the gradients are important for the response one is studying

• How to solve this problem? ‾ Either use compartmental models (still ordinary

differential equations) or create models using partial differential equations.

‾ Effectivelly, PDEs are solved using compartments.

From networks to physiological behavior

Network representations

Types of problems

Typical bottlenecks and assumptions in model building