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Prof:Rui Alves Dept 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? X0X1X2X3 X0 X1 X2 X3 t0t1t2t3

t X3 X0X1X2X3

 Build mathematical models!!!!

 From networks to physiological behavior  Graphical network representations  Types of problems  Typical bottlenecks and assumptions in model building

AB What does this mean? Possibilities: A B Function B A A B A B B A

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 2 3 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 A 3 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 A B D X3 X1 X2 X4

X2 X3 + 2 X1 3 X4 C A B D X3 X1 X2 X4

X2 X0 Production Reaction Sink Reaction

X2 X0 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.

A B C +

C + A or C Flux Linear Saturating 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 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: ; J Bacteriol. 184:  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 )  A regular cell that we represent in a simplified maner  E-cell project represents the E. coli cell using linear kinetics.

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) Steady state value

 Because if biological systems were not robust, we would not be alive, given that fluctuations happen all the time. Parameter (T, kinetic parameters) Steady state value

 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) P1P2 P… Pn M1 M2 M… Mn

How does the specific part of a cell responsible for a given function works???  How does it work qualitatively  Network reconstruction P1 P2 P… Pn M1 M2 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 Proteins 57:481 Vilella et. al Comp. Func. Genomics 5:328 Alves et. al Proteins 56:354 Alves & Sorribas 2007 BMC Systems Biology 1:10 PredictionVerified? Grx5 modulates Nfs1 and Scaffold activity/Interactions Detected interaction with scaffolds Arh1-Yah1 act on S or STYes [ PNAS 97:1050; JBC 276:1503 ] Arh1-Yah1 interaction same as in mammals No reported experiment Yfh1 acts on S, T, or STYes [ Science 3 05:242; 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 RNo reported experiment Chaperones act on Folding, StabilityYes for Folding [ JBC 281:7801 ] Alves et al 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 M1 M2 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]

X0X0 X1X1 _ + X2X2 X3X3 X4X4 X0X0 X1X1 _ + X2X2 X3X3 X4X4 __ 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] Stimulus Overall 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)?

 Individual steps of all processes are mass action  The kinetics of a process may be complicated X1 X2 P1 X1 P1 X1P1X2P1 X2 P1

 We may end up with a model that is larger than it has to be: 2 Variables4 Variables  Use rational enzyme kinetics to reduce model dimension: e.g. HMM kinetics X1 X2 P1 X1 P1 X1P1X2P1 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

 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

 If parameter values are available, then solutions can be numerically calculated (PM)  Numerical solutions allows us to predict the behavior of a specific system

X0X0 X1X1 _ + X2X2 X3X3 X4X4 Constant Protein using X 3

Steady state response Long term or homeostatic systemic behavior of the network

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

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