Simulation algorithms for FRAP experiments with actin system
In our model we use the model of filament structure as presented in Fig. 1.
Filament
Pointed end
konp
Barbed end
koffb
d
koffp
konb
klf krt
lbl
l
Figure 1. Filament structure in actin polymerization model
The filament is presented as the chain of protomers, each protomer could be inbleached
or unbleached state. The distribution of bleached actins along filament immediately after bleach
depends on space and directional distribution of filaments in the investigated region and should
be taken into account as initial conditions in the model. Our goal is to derive model, which will
give correct FRAP recovery for different initial conditions, filament lengths, and reaction rates.
Without loss of community we can assume in particular filament of length l bleached actins form
permanent subchain of of length lbl and and center which is separated from the pointed end by d
actins. All kinds of initial conditions could be obtained by making statistical averaging according
the distribution of l, lbl and d values.
To describe the change of filament structures due to occurring reactions two alternative
methods could be used:

Direct calculation of structure modification due to occurring reactions. It assumes
look on association/dissociation reactions on filaments ends and register the events separately for
bleached and unbleached actins. There are no limits for such approach when we use simulation
modeling. To develop analytical equations superposition of the flows of events should be
considered.

Considering of structure changes as virtual migration of marked actins through the
filament;
Direct calculation of structure modification
Association/dissociation reactions on filaments ends have stochastic nature, i.e. for the
particular system composition and short time step dt probability of i-th reaction occurrence could
be calculated (Gillespie, 1977):
pi  ai dt
(1)
1
where ai is the product of volume- and composition-independent reaction rate ki and
number of combinations of reacting molecules (we consider barbed and pointed ends of
filaments as reagents (Nazarov et al., 2007)).
Under assumption of steady state for the system, which means constant concentration of
actin monomers and filaments and constant mean filament length, ai values do not depend on
time. So flow of events occurrence for i-th reaction could be considered as Poisson flow of
events with intensity ai . Number of occurring events m during finite time t should in compliance
with Poisson distribution. The corresponding probability is calculated by Eq. 2:
pim 
ait m e a t
(2)
i
m!
konb
koffb
konb
koffp
a
b
Figure 2.Two simplified cases of actin polymerization processes:
a.
b.
association and dissociation on barbed end;
association on barbed end and dissociation on pointed end;
Two simplified systems (Fig. 2) are investigated to test the possibilities of this method:
1. Two reactions are considered – the association of actin on barbed end and dissociation of
actin on pointed end (Fig. 2b). The possibility of actin to dissociate from filament is possible
only through dissociation reaction. The probability that actin, initially separated from the
filament pointed end on d monomers is not the protomer of filament after time t is:

d 1
m x
m 0
m
t   1   poffm t 
Px t    poff
(3)
For our tests we assume the same probability to bleach protomers with different positions
according filament barbed and pointed ends. The resulted FRAP curve is the result of averaging
of Px t  values, where x is whole number universally distributed in the range [0,l-1]:
FRAP (t )  1 
1 l 1
 Px (t )
l x 0
(4)
2
The difference of 1 with average probability value is taken, because FRAP signal is
formed by unbleached actins.
2. Association and dissociation reactions on barbed end and inactive pointed end (Fig. 2a).
In this case dissociation of bleached actins occurs through barbed end, which is affected by
association reaction. The dynamics of barbed end is determined by superimposing of two
Poisson flows of events. One of them will describe elongation process, the second will
correspond to dissociation reactions. For determining the probability of some actin with the fixed
initial position to dissociate after certain time, not only number of events, but also their order
should be taken into account. Solving of this problem will further require application of special
methods of the theory of stochastic processes.
As shown in this example, the derivation of analytical equation even in the simple case is
not trivial problem. In the common case with two association and two dissociation reactions the
calculation of probabilities for dissociation of bleached actins is hardly possible.
Approximation of structure modification by virtual diffusion of bleached actins
Looking on virtual diffusion of protomers in the filament is an alternative way to simulate
FRAP experiments with actin polymerization system. It allows to overcome the difficulties of
accounting to the order of different reactions in the direct method. The idea is to approximate the
output of the occurring reactions as virtual migration of bleached actins in a filament. There are
several factors, which should be considered for developing analogs between diffusion and actin
migration processes:

Both diffusion and actin migration processes are stochastic processes with certain
probability for the particle to move on the certain distance during the certain time;

For the unlimited diffusion movement of particle in each direction has the same
probability. Because probabilities of the reactions on the filaments ends are different there is
preferable direction for protomers diffusion in filaments
Conclusion
In our current work we use the second approximative method for the derivation of
analytical model of FRAP experiments with actin polymerization system. The derivation of
analytical equation is given in Supplementary File 2. We used the first direct method to build
computational stochastic simulation algorithm, which operates with ensemble of filament
structures and uses stochastic algorithm to determine time and kind of occurring reactions. This
method does not have any limitations and allows us to evaluate our analytical approximate
model.
3
References
Gillespie, D.T. 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys.
Chem. 81(25):2340-2361.
Nazarov, P.V., E. Friderich, E. Barsukov, A. Halavatyi, E. Ivachkevich, M.M. Yatskou,
andV.V. Apanasovich. 2007. Developing mathematical models, algorithms and programming
tools for analysis of actin based motility National Research Fund (FNR) Luxembourg Final
Report.
4