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Prof:Rui Alves Dept Ciencies Mediques Basiques, 1st Floor, Room 1.08 Website:

Hey, it’s raining!!! Why don’t we try and figure out how all the little molecular pieces in a cell work together and do stuff?!?!?! Look!! We now know how to use bioinformatics to reconstruct biological networks. What now?

Prof:Rui Alves Dept Ciencies Mediques Basiques, 1st Floor, Room

 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?

No

 What do the interactions between nodes mean?  No!!!  Which proteins are important regulatory points in the dynamic responses?

No

 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 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!!!!

 From networks to physiological behavior  Graphical network representations  Mathematical formalisms  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

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 B C 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)

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 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 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 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 2 – Metabolite 2 is produced from metabolite 1 by enzyme 2 3 – Metabolite 3 is produced from metabolite 2 by enzyme 3 4 – Metabolite 4 is produced from metabolite 3 by enzyme 4 5 – Metabolite 5 is produced from metabolite 3 by enzyme 5 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? X0X1X2X3 X0 X1 X2 X3 t0t1t2t3 E1E2E3E4

t X3 X0X1X2X3 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 Chance THE KINETICS OF THE ENZYME-SUBSTRATE COMPOUND OF PEROXIDASE J. Biol. Chem., Dec 1943; 151:

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

A B C +

C + A or C Flux Linear Saturating Sigmoid

A B C + 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)

A B C + f(A,C) can be approximated by considering only a few of its mathematical derivatives with respect to the variables (A,C)

A B 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.

A B C + g<0 inhibits flux g=0 no influence on flux g>0 activates flux Use Taylor theorem in Log space Power 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

X0X1X2 X3 X4 _ _ _ _ + +

 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

X0X1 X2 X3 X4X X6 _ +

 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 C

X0X1 X2 X3 X4X X6 _ +

 Needs more parameters to create determined models  Can not be solved analytically.

 From networks to physiological behavior  Graphical network representations  Mathematical formalisms  Typical 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 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 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: ; 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 among many others).  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?  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) 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

 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. t X

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

How does the specific part of a cell responsible for a given function works???  How does it work quantitatively or can we reproduce the observations?  Parameter estimation when network is known. 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? Fomekong-Nanfack Y, Kaandorp JA, Blom J. Bioinformatics :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?

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  Mathematical formalisms  Typical types of problems  Typical bottlenecks and assumptions in model building

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)?

 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 approximated 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):  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. 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 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.

 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 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.

 From networks to physiological behavior  Network representations  Mathematical models  Types of problems  Typical bottlenecks and assumptions in model building

X0 X1 X2 X4 X3 X5 X X4 X3 X5 X