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!!!!

Organization of the talk From networks to physiological behavior Network representations Mathematical formalisms Studying a mathematical model

Network representation is fundamental for clarity of analysis AB What does this mean? Possibilities: A B Function B A A B A B B A

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

Organization of the talk From networks to physiological behavior Network representations Mathematical formalism Studying a mathematical model

Representing the time behavior of your system A B C +

What is the form of the function? A B C + A or C Flux Linear Saturating Sigmoid

What if the form of the function is unknown? 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)

Are all terms needed? 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)

Linear approximation 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)

What if system is non-linear? Use a first order approximation in a non-linear space.

Logarithmic space is non-linear A B C + 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

Organization of the talk From networks to physiological behavior Network representations Mathematical formalism Studying a mathematical model

A model of a biosynthetic pathway X0X0 X1X1 _ + X2X2 X3X3 X4X4 Constant Protein using X 3

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

How is homeostasis of the flux affected by changes in X 0 ? Log[X 0 ] Log[V] Low g 10 Medium g 10 Large g 10 Increases in X0 always increase the homeostatic values of the flux through the pathway

How is flux affected by changes in demand for X 3 ? Log[X 4 ] Log[V] Large g 13 Medium g 13 Low g 13

How is homeostasis affected by changes in demand for X 3 ? Log[X 4 ] Log[X 3 ] Low g 13 Medium g 13 Large g 13

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 Solve numerically

Specific adaptive response Get parameter values Get concentration values Substitution Solve numerically Time [X 3 ] Change in X 4

General adaptive response Normalize Solve numerically with comprehensive scan of parameter values Time [X 3 ] Increase in X 4 Low g 13 Increasing g 13 Threshold g 13 High g 13 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? X0X0 X1X1 _ + X2X2 X3X3 X4X4 X0X0 X1X1 _ + X2X2 X3X3 X4X4 __ Overall feedback Cascade feedback

Overall feedback improves functionality of the system TimeSpurious stimulation [C] Overall Cascade Proper stimulus Overall Cascade [C] Stimulus Overall 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 From networks to physiological behavior Network representations Mathematical formalism Studying a mathematical model

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) 1 st order 2 nd order i th order i th + j th order

Are all terms needed? 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)

Linear approximation 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)

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? 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,…)