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Prof:Rui Alvesralves@cmb.udl.es

973702406Dept Ciencies Mediques Basiques,

1st Floor, Room 1.08Website:http://web.udl.es/usuaris/pg193845/testsite/

Prof:Rui Alvesralves@cmb.udl.es

0034973702406Dept Ciencies Mediques Basiques,

1st Floor, Room 1.08http://web.udl.es/usuaris/pg193845/testsite/index.htm

http://web.udl.es/usuaris/pg193845/Courses/IGC_2008/

From networks to physiological behavior

Graphical network representations

Mathematical formalisms

Types of problems

Typical bottlenecks and assumptions in model building

You have a gene or process of interest Genes/Proteins do not work alone How does your gene work in its physiological environment? Use different methods and reconstruct the network where the gene is

working

What do the interactions between nodes mean?

What do the interactions between nodes mean? No!!!

Which proteins are important regulatory points in the dynamic responses?

What do the interactions between nodes mean? No!!!

Which proteins are important regulatory points in the dynamic responses? No!!!

All genes that are fundamental for the function of the network?

No, not really, although you can use some combination of centrality measures to figure out a few.

Maybe if network representation has precise and unambiguous meaning we can do it!!!!

Types of problems

What does this mean?Possibilities:

Function

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

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

Full arrow represents a flux between A and B.

Dashed arrow represents modulation of a flux.

A and B – Dependent Variables (Change

over time)

C – Independent variable (constant

value)

Stoichiometric information needs to be included.Dashed arrow represents modulation of a

Reversible Reaction

Stoichiometric information needs to be included.Dashed arrow represents modulation of a flux.

2 A3 D

Having too many names or names that are closely related may 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

+2 X13 X4

Production

Reaction

Sink Reactio

Organel

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 you are consistent and you know exactly what the different elements of a representation mean.

• 1 – Metabolite 1 is produced from metabolite 0 by enzyme 1

• 6 – Metabolites 4 and 5 are consumed outside the system

• 7 – Metabolite 3 inhibits action of enzyme 1

• 8 – Metabolite 4 inhibits enzyme 4 and activates enzyme 5

• 9 – Metabolite 5 inhibits enzyme 5 and activates enzyme 4

• 1 – mRNA is synthesized from nucleotides

• 2 – mRNA is degraded

• 3 – Protein is produced from amino acids

• 4 – Protein is degraded

• 5 – DNA is needed for mRNA synthesis and it transmits information for that synthesis

• 6 – mRNA is needed for protein synthesis it transmits information for that synthesis

• 7 – Protein is a transcription factor that negatively regulates expression of the mRNA

• 7 – Lactose binds the protein reversibly, with a stoichiometry of 1 and creates a form of the protein that

does not bind DNA.

• 1 – 2 step phosphorylation cascade

• 2 – Receptor protein can be in one of two forms depending on a signal S

• 3 – Receptor in active form can phosphorylate a MAPKKK.

• 4 – MAPKKK can be phosphorylated in two different residues; both can be phosphorylated simultaneously or

in sequence

• 5 – MAPKK can be phosphorylated in two different residues; both can be phosphorylated simultaneously or

in sequence

• 6 – Residue 1 of MAPKK can only be phosphorylated if both residues of MAPKKK are phosphorylated

• 7 – Residue 2 of MAPKK can be phosphorylated if one and only one of the residues of MAPKKK are

phosphorylated.

• 8- Only fully phosphorylated MAPKKK can simultaneouslyphosphorylate both MAPKK residues.

Maybe resolving ambiguity in representation is enough to predict

behavior?

X0 X1 X2 X3

t0 t1 t2 t3E1 E2 E3 E4

X0 X1 X2 X3

Unambiguous network representation is not enough to predict dynamic

behavior.

Unambiguous network representations are necessary but not sufficient for proper network analysis.

Why? Non linear behavior of biological systems!

Build mathematical models!!!!

Britton ChanceTHE KINETICS OF THE ENZYME-SUBSTRATE COMPOUND OF

PEROXIDASEJ. Biol. Chem., Dec 1943; 151: 553 - 577

Types of problems

/dA dt

dA dt A f A Cdt

dAdA dt

dA dt Adt

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

int intintint int

2int int

, ,, ,

operating operatingoperatingpo popo operating operating

operating operatingpo pooperating

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

operatingpooperating

d f A CA 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)

intint

operatingpo

operatingpooperating

dAf A C f A C

df A CA A

df A CC C

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

with respect to the variables (A,C)

1 20, ,

dAf A C f A C k A k C

Taylor Theorem:

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

the variables (A,C)

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

1 2, g gdAf A C A C

g<0 inhibits flux

g=0 no influence on flux

g>0 activates flux

Use Taylor theorem in Log spacePower Law Formalism:

Intuitive parameters Simple, yet non-linear Convex representation in

cartesian space Linearizes exponential space

Many biological processes are close to exponential → Linearizes mathematics

Reproduction of observed behavior

Tayloring of numerical methods to specific forms of mathematical equations

X0 X1 X2

Yes: Linlog, Log-Lin, Inverse formalism, SC formalism, etc…

Linlog and log lin are equivalent to the power law formalism

What does the inverse formalism looks like?

Xi Yi=1/Xi Vi(X) Fi(Yi)=1/Vi

This is what the inverse formalism looks like

Yes: Linlog, Log-Lin, Inverse formalism, Saturating Cooperative formalism, etc…

Linlog and log lin are equivalent to the power law formalism

What does the inverse formalism looks like?

What about the SC formalism?

Xi Yi=1/X Vi(X) Fi(Yi)=Log[1/Vi]

May also be very usefull when representing saturable and cooperative

phenomena

Needs more parameters to create determined models

Can not be solved analytically.

Typical types of problems

That depends on the question!!!!

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

The representations we have discussed so far are helpful but they are not everything that is available out there.

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.

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 sub space of solutions for the distribution of fluxes that allows the cell to fulfill 1. (K-cones)

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 regulation.Whole 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 among many others).

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? Model robustness can be used to identify ill

defined parts of the model. 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)

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

Parameter (T, kinetic parameters)

Use sensitivity analysis:

Absolute sensitivities

Relative sensitivities

Stability of a steady state is the notion that after transient fluctuations in the values of the dependent variables, the system will return to the original steady state it was in before the perturbation.

For example, use eigenvalues or Routh criteria

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

Found that the steady state was unstable.

Identified regulatory interactions that stabilized the steady state. Later 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

M2M…

How does it work qualitatively? Network reconstruction.

P2P… Pn

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

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 3 •Alves et al. 2009 BMC Evol. Biol•Moles & Alves 2009 in preparation

How does it work quantitatively or can we reproduce the observations?

Parameter estimation when network is known.

P2P… Pn

• 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?

Fomekong-Nanfack Y, Kaandorp JA, Blom J. Bioinformatics. 2007 15:3356

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 between 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?

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

StimulusOverall

Cascade

Alves & Savageau, 2000, Biophys. J.

Typical types of problems

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

X1P1 X2P1

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

2 Variables 4 Variables

Use approximated kinetics to reduce model dimension: e.g. HMM kinetics.

( , )mV X

f p XK X

X1P1 X2P1

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.

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): Facilitates 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.

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 high accuracy over a

range 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

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 is 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 of continuity 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 (PDEs).

‾ Effectivelly, PDEs are solved using compartments.‾ Alternatively use discrete event modeling.

Network representations

Mathematical models

Types of problems

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