Download Dynamics of reaction-diffusion systems in non

Dynamics of reaction-diffusion
systems in non-homogeneous media
Paola Lecca, Lorenzo Demattè and Corrado Priami
The Microsoft Research – University of Trento
Centre for Computational and Systems Biology
Objective
To model the dynamics of a stochastic reaction-diffusion system in a nonhomogeneous medium.
A possible way to achieve this goal is to incorporate a model of diffusion
into a biochemical stochastic simulation algorithm, as Gillespie algorithm.
At the mesoscopic intra-cellular scale the parameters governing the
kinetics of the diffusion have to be dependent on local variables, as solute
concentration and frictional forces.
Modelling reaction-diffusion systems
A model of a reaction-diffusion systems consists of two parts
o
a set of biochemical reactions which produce, transform or remove
chemical species
o
a mathematical description of the diffusion process
The great majority of mesoscopic reaction-diffusion models of
intracellular kinetics usually assume that the diffusion coefficient is
constant over the time and the diffusion is so fast that all concentrations
are maintained homogeneous in space.
However, recent experimental data on intracellular diffusion constants
show that this supposition is not necessarily valid even for small
prokaryotic cells.
Spatial effects
Spatial effects are present in many biological systems.
Some examples:
o mRNA movement within the cytoplasm
o Ash1 mRNA localization in budding yeast
o morphogen gradients across egg-polarity genes
Our description of diffusion processes in a highly structured and nonhomogenous medium starts from a generalization of the Fick’s law.
Generalized Fick’s law (1)
The driving force leading to diffusion is the Gibbs energy difference between
regions of different concentrations.
The chemical potential
of a species is defined as the partial derivative of
the Gibbs energy
with respect to the concentration of the species
where
is the standard chemical potential, is the ideal gas constant,
the absolute temperature and
the chemical activity of the species .
is the activity coefficient and
is a reference concentration.
Generalized Fick’s law (2)
The flux defined as the number of number of moles of solute which pass through
a small surface per unit time per unit area is
where
the diffusion coefficient given by
Supposing that the chemical activity of the solute only weakly depends on the
concentration of the other solutes we can assume that
.
Generalized Fick’s law (3)
In 1D- space, the rate of change of concentration of species
is given by
where
and
due to diffusion
and
denote the concentration of the substance
.
at coordinate
, and
Generalized Fick’s law (4)
The rate of diffusion of a substance
at the mesh point
where, by introducing the friction coefficient of species
,
, we have that
(Chemical activity coefficient)
is
at mesh point
(Second virial coefficient)
,
A case study: chaperone-assisted
protein folding
Protein folding, chaperone binding, and misfolded protein accumulation
take place inhomogeneously in the space.
The spatial distribution of chaperones in the cytoplasm may not be
uniform, and consequently the distribution of correct and faulty proteins
may be not uniform. In turn, the time evolution of spatial distribution of
chaperones may affect the time evolution of the spatial distribution of
faulty proteins.
A view of the system
In a 2D space, for simplicity.
Reaction event
Diffusion event
is the diffusion coefficient
of species i at time t = 0.
is the number of species
that diffuse
is the waiting reaction time
is the number of diffusion
events
is the number of reaction
events
At time t, the event
(diffusion or reaction)
that succeds to occur is
the fastest one (the most
probable), as in a
Gillespie-like approach.
Most probable
diffusion direction
Parameters of simulation
Simulation space: a grid of
thus consisting of 81 cells
(each cell has size
nm) is considered.
A 2D diffusion model is simulated and a spatially homogeneous
distribution of nascent_protein and an initial null concentration of
right_protein in every cell are assumed. The density (expressed in
number of molecules per
).
Total simulation time:
Simulation results (1)
Time evolution of chaperone
distribution
Simulation results (2)
Time evolution of ”misfolded_1”
protein distribution
Simulation results (3)
Time evolution of ”misfolded_2”
protein distribution
Simulation results (4)
Time evolution of distribution of
correctly folded proteins.
Spatial average correlation
The spatial correlation between the
proteins and chaperones has been
monitored in terms of the quantity
and
are the concentration of
proteins and chaperones, respectively.
Correlation matrices (1)
Matrices of correlation between
chaperone and “misfolded 1”
protein concentration.
Correlation matrices (2)
Matrices of correlation between
chaperone and “misfolded 2”
protein concentration.
Correlation matrices (3)
Matrices of correlation between
chaperone and correctly folded
protein concentration.
Conclusions
In non-homogenoues media, constant diffusion coefficients w. r. t. the
time and space are rather more than exception than a rule in living
cells.
This work provides a theoretical derivation of the molecular origin of
these parameters.
The results obtained with the simulation of the process of chaperoneassisted protein folding are in agreement with the qualitative and
quantitative experimental data (A. R. Kinjo and S. Takada, Biophys.
Journal, 85:3521-3531.).