Mathematical Biology: The Virtual Cell
– Assignment 1 –
Handed out: Monday, 14th March 2016
Deadline: Monday, 4th April 2016, 9.00h (start of lecture)
• The answers to the exercises must be handed in before start of the lecture.
Make sure these are easily readable, either handwritten or printed.
• Provide sufficient details of the arguments, such that these can be properly
understood.
• Provide references if you use additional sources from the literature.
• Exercise 5 (marked with ∗), must be made by students in Mathematics.
Others may make this exercise, which then counts as a bonus.
1.) By means of the patch clamp measuring technique one is able to measure
the ionic current through a single Na+ -selective ion channel that is located
in a tiny ‘patch’ of the membrane that has been sucked out by means of a
micropipette. In such an experiment one may measure a current through
an open channel of −6.6 pA, when the cross membrane potential is −140
mV. The channel is open, on average, during a time period of about 6 ms.
(a) Compute the number of ions that pass the channel per second.
(b) Assuming that the ions pass the channel sequentially, one at a time,
how long does it take a single ion to cross the 5 nm long channel
on average? Hence, what is their average velocity when passing the
channel?
(c) How many ions pass the single channel on average, when open?
(d) Compute the conductivity of the single open channel, given that the
Nernst potential is +56 mV.
2.) (Exercise 2, p.51) Assume that a proto-cell has a membrane capacitance of
1 µF/cm2 and contains only K+ selective channels. The initial intra- and
extracellular concentrations for K+ are 400 mM and 20 mM respectively,
when all channels are closed. Assume further that temperature is fixed
at 37◦ C and that the positive charge of the K+ ions is balanced by intraand extracellular anions A− equal in concentration to K+ .
(a) Compute the Nernst potential for the concentration distribtution as
indicated.
(b) How many K+ ions must move out of the cell per µm2 to achieve
a voltage difference over the membrane equal to that computed in
part (a)?
(c) Derive a formula for the final intracellular K+ concentration [K+ ]in
as function of cell diameter d (in µm), assuming that our proto-cell
is perfectly round and that the extracellular K+ concentration does
not change (due to experimental set-up). Use the number of ions
transferred that you calculated in part (b). Plot [K+ ]in (in mM) as a
function of the diameter d (in µm) for a range of biologically realistic
diameter values.
(d) Assuming that the extracellular K+ concentration remains constant,
compute the Nernst potential for the range of intracellular K+ concentrations calculated in part (c).
(e) Argue whether it is reasonable or not to use the initial concentrations and the Nernst equation to estimate the final cross-membrane
potential after opening of the channels.
3.) (Based on Excercise 5, p. 51) The timescale τ (V ) for the equilibration of
the fraction of open channels for a voltage-gated ion channel as function
of membrane potential is postulated to be of the form
τ (V ) =
k0+ e−αV
1
+ k0− e−βV
(1)
(cf. equation (2.18)), where k0± > 0. Show that (1) can be rewritten into
the form
eV (α+β)/2
(2)
τ (V ) = q
.
−V0
2 k0+ k0− cosh V2S
0
4.) (Interpretation of voltage clamp measurements). In Figure 1 and Table
1 the outcome of a (virtual) voltage clamp experiment is given as would
be read-off from the voltage clamp device. At time t0 = 5 ms the cross
membrane potential is switched to the indicated level and reset to the
reversal potential at t1 = 25 ms. The ‘experimental set-up’ was made in
such a way that only a single type of ion channel could open. The other
type of ion channels remain closed.
(a) Use the dynamical behaviour of the system as represented in Figure
1 to complete Table 1 for the values of the clamp voltages Vclamp
mentioned in Figure 1. What is the expected range of possible values
for the reversal potential VR ?
(b) Make a detailed and careful plot of the steady-state applied current
∗
against V = Vclamp .
Iapp
(c) Estimate VR from the I, V -plot made in (b). Motivate your estimation technique.
(d) Estimate from (b) the maximal membrane conductance ḡ for this
type of channel. Motivate your estimation technique.
Figure 1: The applied current as function of time when at t = 5 ms clamp potential
is switched from reversal potential VR to indicated clamp potential. At t = 25 ms the
clamp potential is reset to VR .
(e) For each membrane potential V mentioned in Table 1 and Figure
1, compute the steady state fraction of open ion channels f∞ (V ) at
potential V . Motivate your computation and present the results in
a table.
∗
Iapp
(pA)
-0.59
232
351
455
Vclamp (mV)
-90
-20
-10
0
Vclamp (mV)
+10
+20
+30
+40
∗
Iapp
(pA)
553
640
716
795
Table 1: Steady-state applied currents for additional cross-membrane potentials
The steady state fraction of open ion channels f∞ (V ) at potential V is
given by the expression
f∞ (V ) =
1
2
1 + tanh
V − V0 1
.
=
0
2S0
1 + exp − V S−V
0
(f) Use your results from (e) to plot
f∞ (V )
y := ln
1 − f∞ (V )
(3)
as function of V . Then use this plot to estimate V0 and S0 . Motivate
your estimation procedure.
(g) Derive that the graph of Iapp (t) for t0 ≤ t ≤ t1 can be described by
∗
Iapp (t) = Iapp
− Ce−(t−t0 )/τ ,
(4)
∗
where C, τ > 0 and Iapp
represents the (asymptotically) maximum
value of Iapp .
(h) Use (4) and Figure 1 to plot an appropriate function of Iapp against
time t in order to obtain an estimate for τ (V ) in case V = −25 mV.
5.)∗ (Students in Mathematics – or bonus) Reconsider the switching of an ion
channel between a closed and open state (C and O) as discussed in the
lectures: the time T+ till next transition from closed to open and T−
of open to closed are exponentially distributed with rates λ+ and λ−
respectively.
(a) Prove that the probability of the occurrence of precisely two transitions (take e.g. C → O → C) within the time interval [t, t + ∆t] is
of order ∆t2 .
The probability of the occurrence of precisely n transitions in the time
interval [t, t + ∆t] is of order ∆tn .
(b) Let p(t) be the probability that a single ion channel as described
above is in open state at time t. Assuming that t 7→ p(t) is differentiable, prove that p(t) satisfies the differential equation
dp
(t) = λ+ 1 − p(t) − λ− p(t).
dt
(5)
Points per exercise:
1.
8
2.
11
3.
3
(4 × 2)
(2 + 2 + 3 + 2 + 2)
4.
23
5. (for Math students)
9
(3 + 2 + 2 + 2 + 3 + 4 + 4 + 3)
(5 + 4)
Grading:
Math students =
Non − math students =
1
6 6
1
5 5
+ total number of points ,
+ total number of points ,