Menko Ypma
BIOEN 498 Quantitative Physiology
2/21/15
Large Question (5-7.4 page 85)
Problem: Write a set of partial differential equations for axonal transport of vesicles of
acetylcholine being carried to a synaptic junction at the end of a motor nerve. Given a free
diffusion coefficient for vesicles of 0.1 μm2 s-1 , a rate of kinesin binding to the vesicle of 2 uM1 -1
s and a release rate of 0.5 s-1 , a rate of cargo-loaded kinesin binding to the tubulin
microfilament of 1 s-1 and a release rate of 0.1 s-1 , and a forward velocity of 0.5 μm s-1 for kinesin
along the microtubule: (1) How many ATPs are used per μm? (2) Given a square wave labeling
of vesicles from x = 0 to x = 100 μm at t = 0, what are the concentration profiles for bound and
for freevesicles at 1, 10 and 100 seconds? After 10 seconds, what fraction of vesicles are linked
to microtubules and what fraction are diffusing, with or without kinesin attached? (4) How much
ATP was used by 100 seconds? (5) Given that no taxol was used, what was the average number
of microtubules encountered by a vesicle being carried 100 μm?
The model:
A Jsim model was constructed (see code). The model consisted of a single chamber in which
three species (free vesicle, vesicle bound to kinesin, and vesicle bound to kinesin and
microtubule) were modeled using bulk concentrations. These are labled as Cv, Cvk, and Cvkt
respectively. Reaction rates and bulk diffusion coefficients were defined as given in the
problem. Unbinding from microtubules due to degradation of the microtubules was modeled as
a first order irreversible reaction with a time constant of 1/600seconds (1/10minutes).
Concentrations of ATP, kinesin, and microtubules were assumed to be constant throughout
both space and time and so are not included in the differential equations (including them would
conflict with the units of the reaction rates given). Diffusion coefficients were assumed to be
identical for vesicles with and without kinesin. Movement along the microtubules was
approximated as a product of the velocity and the space derivative of concentration. The
chamber was arbitrarily given a length of 200um and a cross sectional area of 1um for purposes
of scaling total amounts of certain species or events. Initial conditions were defined as a
distribution of labeled vesicles in space, with none bound to either kinesin or microtubules at
the beginning.
1. “How many ATPs are used per um?”
a. If 1 ATP is required to move the vesicle the distance on 1 dimer (8nm=.008um)
1(𝑢𝑚)
𝐴𝑇𝑃
then the number of ATP required for 1um is 1 (𝐴𝑇𝑃)/.008(𝑢𝑚) = 125 ( 𝑢𝑚 )
2. “Given a square wave labelling of vesicles…”
a. Given the above model with a square wave input in space (defined as Pulse 1
with start 5, duration 100, amplitude 1) the concentration profiles at different
times are shown in Figure 1. Note that, over time the concentration profile shifts
to the right (in the direction of kinesin movement) and broadens somewhat due
to diffusion in the unbound states. By t=10 the total concentrations of each
species appear to have reached equilibrium (see Figure 2).
3. “Given the above, after 10 seconds, what fraction of vesicles are linked to
microtubules…”
a. As shown in Figure 2 the total concentrations of each species are approximately
at equilibrium by 10 seconds. At 10 seconds approximately 2.2% of vesicles are
completely free, 9.1% are free with kinesin bound, and 88.7% are linked to
microtubules.
4. “How much ATP was used by 100 seconds”
a. Given the above parameters, and assuming that ATP use is proportional to the
amount of species vkt (vesicle linked to microtubule), with a constant of
proportionality defined based on system parameters as follows:
ATP
um
s
125 (
) ∗ 0.5 ( ) ∗ .1(
)
um
s
timestep
This assumes that vesicles which start bound to microtubules all consume move
over the entire time step, in reality some vesicles will unbind during a timestep
and consume less ATP. However as the length of each timestep and the
unbinding rate are both quite small, this is probably negligible. By 100 seconds a
total of 5.457E-11 mmol of ATP was used for the transport of 1E-13 mmol of
vesicle. This comes out to about 545 molecules of ATP per vesicle per 100
seconds.
5. “What was the average number of microtubules encountered…”
a. Given a simulation length of 225sec, in which the mean of the concentration
profile moves approximately 100um (see Figure 3) the number of distinct
microtubule binding events is proportional to the integral of the binding rate
multiplied by concentration, over both time and space. Dividing this number by
the total number of vesicles gives an average of 23.29 binding events for a
vesicle traveling 100um.
Figure 1:Concentration Profiles at 1,10 and 100 seconds for bound (Cvkt) and free (Cv+Cvk) vesicles (plotpage_6 in model).
Figure 2: Total amounts of each species in the chamber over time. Y axis is in units of mmol assuming a
cross sectional area of (1um^2) and length of 200um. The black line is the total concentration of all
species and demonstrates that the simulation has a mass balance. (plotpage_4 in model)
Jsim Code :
// ---------------------------------------------------------------------------// ----------------------------------------------------------------------------
import nsrunit;
unit conversion on;
math Vesicles{
// ---------------------------------------------------------------------------//
PARAMETERS
// ----------------------------------------------------------------------------
real
Dv = 0.01 um^2/sec,
//
// MODEL PARAMETERS
// Diffusion coefficient
k are rate constants to switch between bound states of the
vesicle
k1
= 2 1/sec, //rate for conversion of free vesicle to
vesicle bound to kinesin
k1r
= 0.5 1/sec, //rate for conversion of vesicle bound
to kinesin to free vesicle
k2
= 1 1/sec, //rate for conversion of vesicle bound to
kinesin to vesicle bound to kinesin and tubulin
k2r
= 0.1 1/sec, //rate for conversion of vesicle bound
to kinesin and tubulin to vesicle bound to kinesin
r_vkt
= 0.5 um/sec, //rate of advancement of vk along a
microtubule
k_tubule =1/600 1/sec, //effective rate of unbinding to
tubule due to disassembly
ATPperum = 125 1/um,
//molecules of ATP needed to move 1
vesicle 1um
CellXArea = 1 um^2; //cross sectional area, chosen
arbitrarily
// ---------------------------------------------------------------------------//
VARIABLES
// ----------------------------------------------------------------------------
realDomain
// DOMAIN VARIABLES
t sec; t.min=0;
// Time domain
t.max=20.0; t.delta=0.016;
realDomain
x um; x.min=0;
// Spatial domain
x.max=10 ; x.delta=.008 ;
real
// MODEL VARIABLES
Cv(x,t) mM,
// Concentrations of species
in region
Cvk(x,t) mM,
Cvkt(x,t) mM,
ATPuse(x,t) mM,
// ATPusage as a
function of time and space
IntATP(t) mmol,
// Integral over space
of ATPusage
TotalATPuse mmol*sec,
//Integral over time of
IntATP, total ATP used over simulation
TubEnc(x,t) mM/sec,
//vesicle-
microtubule encounters as a function of time,space
TotTubEnc mmol,
//integral of
encounter over all time and space
IntCv(t) mmol,
Cv as a function of time
//total amount of
IntCvk(t) mmol,
//total amount of
Cvk as a function of time
IntCvkt(t) mmol,
//total amount of
Cvkt as a function of time
IntTot(t) mmol;
//total amount of
vesicular species as a function of time
extern real Cvin(x) mM;
// ---------------------------------------------------------------------------//
INITIAL CONDITIONS
//
Assume everything is 0 to begin, with a set profile of Cv
in space
//
Problems arise of Cvkt(0,t) is greater than 0, shift
starting distribution to avoid this.
//
IntTot should be constant over all time.
// ----------------------------------------------------------------------------
when (t=t.min) {
Cv
= Cvin;
Cvk
= 0;
Cvkt = 0;
}
when (x=x.min) {
Cv:x
= 0;
Cvk:x
= 0;
Cvkt:x = 0;
}
when (x=x.max) {
Cv:x
= 0;
Cvk:x
= 0;
Cvkt:x = 0;
}
// ---------------------------------------------------------------------------//
SYSTEM OF EQNS
// ----------------------------------------------------------------------------
//
Governing PDES
Cv:t
= Dv * (Cv:x:x)
- k1*Cv + k1r*Cvk;
Cvk:t = Dv * (Cvk:x:x) + k1*Cv - k1r*Cvk - k2*Cvk +
k2r*Cvkt + k_tubule*Cvkt;
Cvkt:t = -r_vkt*(Cvkt:x) + k2*Cvk - k2r*Cvkt k_tubule*Cvkt;
//
Additional Calcs
ATPuse= Cvkt*ATPperum*r_vkt*t.delta;
IntATP=
integral(x=x.min to x.max, ATPuse)*CellXArea;
TotalATPuse=integral(t=t.min to t.max, IntATP);
TubEnc=Cvk*k2;
TotTubEnc=integral(t=t.min to t.max, integral(x=x.min to
x.max, TubEnc)*CellXArea);
IntCv = integral(x=x.min to x.max, Cv)*CellXArea;
IntCvk = integral(x=x.min to x.max, Cvk)*CellXArea;
IntCvkt = integral(x=x.min to x.max, Cvkt)*CellXArea;
IntTot = IntCv+IntCvk+IntCvkt;
}