Mathematical Comments on
Systems Biology: examples
and questions
Elisabeth Pécou
Institut de Mathématiques de Bourgogne (U. de Bourgogne)
and
Centro 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.
SWITCH
SHIFT
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
Cuint
Cuext
• Mathematical homeostasis condition is satisfied.
• The stationary state does not depend on the
source of copper.
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 ».