Mathematical Comments on Systems Biology: examples and

questions

Elisabeth PécouInstitut de Mathématiques de Bourgogne (U. de Bourgogne)

andCentro de Modelamiento Matemático (U. de Chile)

Biomolecular Systems

• Genes, proteins, metabolites,… • Activation, inhibition, chemical

transformations, signal transduction,… INTERACTION GRAPH (gene

networks, metabolic pathways,…)• Adaptability, evolvability, robustness,

social behaviour,…

Example: Morphogenetic movments in Dictyostelium

amoebae

• A few cells start to produce periodic cAMP pulses which are detected, amplified and relayed by surrounding cells, forming spiral waves.

• Dictyostelium cells move by chemotaxis toward increasing cAMP concentration and then aggregate to form a mound.

Copyright, M.J. Grimson & R.L. Blanton

Systems Biology Theory

• Understand the emerging properties, like self-organization, of complex biomolecular systems.

A fundamental system behaviour: switching

• Switching implies that the underlying dynamical systems has two stationary stable states.

Interaction graph

Thomas-Soulé Theorem

• If a system admits more than one non degenerate stationary states on an open domain, then at least one of its local interaction graphs has a positive circuit.

• Example: in the domain ]0,+[x]0,+ [x]r,+[, the Lorenz system has at most one stationary state.

Switching occurs by bifurcation

• Discontinuous experimental « curve »

• Equilibrium curve: the 0-level curve of the vector-field

• pc is a saddle-node bifurcation point

Hysteresis effect:  « irreversibility »

• « Irreversibility »: when the parameter decreases under pc

no jump occurs.

• This criterion distinguishes switch phenomena from simple shifts.

SHIFT

SWITCH

Example: Genetic Toggle Switch

(Gardner, Cantor, Collins)

• Synthetic gene regulatory network of two genes, negatively interacting on each others (cooperative repression) which exhibits bistability for a wide range of parameter values.

Perspectives

• Bistability is at the basis of cell differenciation or cancer proliferation. Work remains to be done to understand how multistability is effectively achieved in those cases and how to control the process at the system level.

• Successful construction of synthetic genetic networks open the doors to an infinite range of applications in therapeutic strategies and in industrial processes.

Homeostasis, another fundamental system

behaviour

• A biological system is open. It exchanges matter, energy and information with its environment.

• Homeostasis is the capacity of a system to maintain constancy even in the face of changing environment (Claude Bernard, Walter Cannon).

Mathematical homeostasis

• Existence of a stable equilibrium state or a stable periodic orbit with small amplitude.

• Negative loop in the interaction graph: necessary condition for homeostasis.

• Weakness: the definition does not take into consideration changes in the environment.

Copper homeostasis in Enterococcus hirae*

• Cu essential micronutrient, toxic at high concentration.

Necessity of a tight regulation.

• Homeostasis of Cu in E. hirae involves the cop operon (see M.Solioz and coworkers).

*joint work with A. Maass, D. Remenik, J. Briche, and M. Gonzalez (2005)

cop operon of E. hirae

The biological model

• Virtual variable (Cu)int;

• Assume all copper needs are fulfilled

• Thus, the criterion for homeostasis reads:

(Cu)int 0 (at equilibrium)

The mathematical model

• Model of a population of identical cells.• Mass-action kinetics, except passive

uptake and outake, modelled by threshold functions (either non linear sigmoidal, or piecewise constant).

• Four classes of models, according to: Expelled copper is immediatly recycled or not; Timescales are taken onto account either by a

time delay in transcription or by the introducing mRNAs as variables.

The equations

Results: Existence of equilibrium

• Theorem: for any values of the parameters,and any external point source of copper, the system admits an equilibrium state for which (Cu)int=0

• Mathematical homeostasis condition is satisfied.• The stationary state does not depend on the source of copper.

Cuint

Cuext

SBMLSim: Simulation tool

• SBMLSim: MATLAB program which presents a graphical user interface, solving and plotting the solutions of the differential system. It allows the user to easily change the parameters and initial conditions.

• The input are standard SBML files which can be generated by Cell designer.

• http://www.dim.uchile.cl/SBMLSim

Transient dynamics

• Internal copper reaches a maximum and then decreases toward 0.

• The parameters which influence the maximum value are the uptake velocity and the transcription velocity.

Changing kA

Changing CA

Sources

• On Dictyostelium• S. Sawai, P.Thomason, E. Cox, Nature 433 (2005)• E. Palsson, E. Cox, P.N.A.S. 93(1997)

• On bistability and positive feedback loops• C. Soulé, ComplexUs 1 (2003)• J.Monod, F. Jacob, Cold Spring Harbor Symp Quant. Biol. 26

(1961).• J. Ferrell, Cur. Op.Chem.Biol. 6 (2002).

• The genetic toggle switch• T. Gardner, C. Cantor, J. Collins, Nature 403 (2000)

• Homeostasis• E. Pécou, A. Maass, D. Remenik, J. Briche, M. Gonzalez

(2005) « A mathematical model for copper homeostasis in Enterococcus Hirae »

• D. remenik, A. Maass, E. Pécou, J. Briche, M. Gonzalez (2005) « SBMLSim: a Matlab GUI for simulating SBML models ».