Numerical analysis of Ca2+ depletion in the transverse tubular system of mammalian muscle.
ABSTRACT Calcium currents were recorded in contracting and actively shortening mammalian muscle fibers. In order to characterize the influence of extracellular calcium concentration changes in the small unstirred lumina of the transverse tubular system (TTS) on the time course of the slow L-type calcium current (I(Ca)), we have combined experimental measurements of I(Ca) with quantitative numerical simulations of Ca2+ depletion. I(Ca) was recorded both in calcium-buffered and unbuffered external solutions using the two-microelectrode voltage clamp technique (2-MVC) on short murine toe muscle fibers. A simulation program based on a distributed TTS model was used to calculate the effect of ion depletion in the TTS. The experimental data obtained in a solution where ion depletion is suppressed by a high amount of a calcium buffering agent were used as input data for the simulation. The simulation output was then compared with experimental data from the same fiber obtained in unbuffered solution. Taking this approach, we could quantitatively show that the calculated Ca2+ depletion in the transverse tubular system of contracting mammalian muscle fibers significantly affects the time-dependent decline of Ca2+ currents. From our findings, we conclude that ion depletion in the tubular system may be one of the major effects for the I(Ca) decline measured in isotonic physiological solution under voltage clamp conditions.
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ABSTRACT: Caveolins are plasma-membrane-associated proteins potentially involved in a variety of signalling pathways. Different mutations in CAV3, the gene encoding for the muscle-specific isoform caveolin-3 (Cav-3), lead to muscle diseases, but the underlying molecular mechanisms remain largely unknown. Here, we explored the functional consequences of a Cav-3 mutation (P104L) inducing the 1C type limb-girdle muscular dystrophy (LGMD 1C) in human on intracellular Ca2+ regulation of adult skeletal muscle fibres. A YFP-tagged human Cav-3P104L mutant was expressed in vivo in muscle fibres from mouse. Western blot analysis revealed that expression of this mutant led to an ∼80% drop of the level of endogenous Cav-3. The L-type Ca2+ current density was found largely reduced in fibres expressing the Cav-3P104L mutant, with no change in the voltage dependence of activation and inactivation. Interestingly, the maximal density of intramembrane charge movement was unaltered in the Cav-3P104L-expressing fibres, suggesting no change in the total amount of functional voltage-sensing dihydropyridine receptors (DHPRs). Also, there was no obvious alteration in the properties of voltage-activated Ca2+ transients in the Cav-3P104L-expressing fibres. Although the actual role of the Ca2+ channel function of the DHPR is not clearly established in adult skeletal muscle, its specific alteration by the Cav-3P104L mutant suggests that it may be involved in the physiopathology of LGMD 1C.Pflügers Archiv - European Journal of Physiology 11/2008; 457(2):361-375. · 4.87 Impact Factor
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ABSTRACT: The aim of this chapter is to describe and discuss a fundamental concept for studying cellular signalling. It is the combination of state of the art experimental techniques, in particular high resolution fluorescence imaging, with detailed spatio-temporal mathematical models of intracellular calcium regulation. This approach provides a powerful tool to elucidate the very complex mechanisms involved in cellular Ca 2+-signalling and allows to correct for inherent experimental limitations and biases. It has proved to be particularly successful, since experimental findings can be directly incorporated into mathematical models. In principle, this allows to use the most accurate assumptions and parameters for the model calculations and vice versa, model predictions can be directly tested in experiments.12/2003: pages 201-230;
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ABSTRACT: A two-microelectrode voltage clamp and optical measurements of membrane potential changes at the transverse tubular system (TTS) were used to characterize delayed rectifier K currents (IK(V)) in murine muscle fibers stained with the potentiometric dye di-8-ANEPPS. In intact fibers, IK(V) displays the canonical hallmarks of K(V) channels: voltage-dependent delayed activation and decay in time. The voltage dependence of the peak conductance (gK(V)) was only accounted for by double Boltzmann fits, suggesting at least two channel contributions to IK(V). Osmotically treated fibers showed significant disconnection of the TTS and displayed smaller IK(V), but with similar voltage dependence and time decays to intact fibers. This suggests that inactivation may be responsible for most of the decay in IK(V) records. A two-channel model that faithfully simulates IK(V) records in osmotically treated fibers comprises a low threshold and steeply voltage-dependent channel (channel A), which contributes ∼31% of gK(V), and a more abundant high threshold channel (channel B), with shallower voltage dependence. Significant expression of the IK(V)1.4 and IK(V)3.4 channels was demonstrated by immunoblotting. Rectangular depolarizing pulses elicited step-like di-8-ANEPPS transients in intact fibers rendered electrically passive. In contrast, activation of IK(V) resulted in time- and voltage-dependent attenuations in optical transients that coincided in time with the peaks of IK(V) records. Normalized peak attenuations showed the same voltage dependence as peak IK(V) plots. A radial cable model including channels A and B and K diffusion in the TTS was used to simulate IK(V) and average TTS voltage changes. Model predictions and experimental data were compared to determine what fraction of gK(V) in the TTS accounted simultaneously for the electrical and optical data. Best predictions suggest that K(V) channels are approximately equally distributed in the sarcolemma and TTS membranes; under these conditions, >70% of IK(V) arises from the TTS.The Journal of General Physiology 08/2012; 140(2):109-37. · 4.73 Impact Factor
Numerical Analysis of Ca2?Depletion in the Transverse Tubular System
of Mammalian Muscle
Oliver Friedrich,* Thomas Ehmer,* Dietmar Uttenweiler,*†Martin Vogel,* Peter H. Barry,‡and
Rainer H. A. Fink*
*Institute of Physiology and Pathophysiology, Medical Biophysics, University of Heidelberg, INF 326, D-69120 Heidelberg, Germany;
†Interdisciplinary Center for Scientific Computing, University of Heidelberg, INF 368, D-69120 Heidelberg, Germany; and‡School of
Physiology and Pharmacology, University of New South Wales, Sydney 2052, Australia
characterize the influence of extracellular calcium concentration changes in the small unstirred lumina of the transverse
tubular system (TTS) on the time course of the slow L-type calcium current (ICa), we have combined experimental measure-
ments of ICawith quantitative numerical simulations of Ca2?depletion. ICawas recorded both in calcium-buffered and
unbuffered external solutions using the two-microelectrode voltage clamp technique (2-MVC) on short murine toe muscle
fibers. A simulation program based on a distributed TTS model was used to calculate the effect of ion depletion in the TTS.
The experimental data obtained in a solution where ion depletion is suppressed by a high amount of a calcium buffering agent
were used as input data for the simulation. The simulation output was then compared with experimental data from the same
fiber obtained in unbuffered solution. Taking this approach, we could quantitatively show that the calculated Ca2?depletion
in the transverse tubular system of contracting mammalian muscle fibers significantly affects the time-dependent decline of
Ca2?currents. From our findings, we conclude that ion depletion in the tubular system may be one of the major effects for
the ICadecline measured in isotonic physiological solution under voltage clamp conditions.
Calcium currents were recorded in contracting and actively shortening mammalian muscle fibers. In order to
One of the main problems in interpreting voltage clamp data
in a variety of biological cells including skeletal and heart
muscle is the current flow-induced accumulation or deple-
tion of ions due to concentration changes in diffusion re-
stricted near membrane locations, such as in the transverse
tubular system (TTS) of skeletal muscle. The TTS consists
of a mesh of small tubules as invaginations from the surface
membrane and composes ?0.3–0.4% of the fiber volume in
frog sartorius muscle (Peachey, 1965) and 0.5–0.6% in rat
laryngeal and sternomastoid muscle (Dulhunty, 1982; Hin-
richsen and Dulhunty, 1982) and spans throughout the mus-
According to the Nernst-Planck equation, ion concentra-
tion changes such as Ca2?depletion result in reduced driv-
ing forces and therefore influence the time course of re-
corded currents. Ca2?depletion in the TTS has been shown
to be a predominant factor in the decay of the slow L-type
Ca2?current (ICa) in frog skeletal muscle fibers using the
Vaseline gap technique (Almers et al., 1981). However,
under different experimental conditions, Cota et al. (1984)
proposed that ICadecay was due to a voltage-dependent
inactivation process in the first place and not due to deple-
tion of tubular Ca2?. With the 3-MVC (three-microelec-
trode voltage clamp technique) in intact fibers under hyper-
tonic conditions, the decay of ICain amphibian skeletal
muscle was explained exclusively by a voltage-dependent
inactivation mechanism (Cota and Stefani, 1989). However,
there is also evidence that tubular Ca2?depletion exists in
fibers bathed in isotonic solutions (Almers et al., 1981;
Lorkovic and Ru ¨del, 1983; Friedrich et al., 1999) and in
stretched fibers (Cota et al., 1984; Miledi et al., 1983;
Nicola-Siri et al., 1980).
Furthermore, previous investigations and simulations of
the influence of concentration changes on the calcium cur-
rents (ICa) did not sufficiently include potentially important
properties of the calcium channel; for example, the voltage
dependence and the intrinsic time course of inactivation
(Levis et al., 1983). Thus, the role of Ca2?depletion in the
TTS in skeletal muscle is still controversially discussed
(Beaty and Stefani, 1976; Stanfield, 1977; Sa ´nchez and
Stefani, 1978, 1983; Almers et al., 1981; Almers and
Palade, 1981; Cota et al., 1983; Francini and Stefani, 1989;
Francini et al., 1992; Garcı ´a et al., 1992).
The aim of this study was to investigate the effect of
calcium ion depletion on ICain skeletal muscle including the
above-mentioned intrinsic channel parameters. We there-
fore present a method where an experimentally recorded ICa
trace serves as input data for a numerical simulation. The
numerical approach was necessary, as the analytical analy-
sis of ion concentration changes and the electrical responses
is, in general, not practicable due to the nonlinearity of the
underlying processes (Eisenberg et al., 1977; see also Ut-
tenweiler and Fink, 1999). In the first set of our recorded ICa
traces (input data for the simulation) tubular Ca2?ion
depletion was impaired by an extracellular buffering agent.
From these input data the effect of Ca2?depletion on the
Received for publication 8 March 2000 and in final form 31 January 2001.
Address reprint requests to Prof. Dr. Rainer H. A. Fink, Institute of
Physiology and Pathophysiology, Medical Biophysics, Im Neuenheimer
Feld 326, D-69120 Heidelberg, Germany. Tel.: ?49-(0)-6221-544065;
Fax: ?49-(0)-6221-544123; E-mail: firstname.lastname@example.org.
© 2001 by the Biophysical Society
2046Biophysical Journal Volume 80May 20012046–2055
time course of ICain unbuffered solution was calculated.
The resulting output was then compared with the experi-
mentally recorded Ca2?currents from the same fiber in
unbuffered solution, which did not contain the calcium
buffering agent, and thereby allowing Ca2?depletion to
occur. Such an approach should also prove most helpful in
clarifying the contribution of Ca2?depletion when record-
ing currents in other tissues, e.g., in cardiac and nerve cells
(Amsellem et al., 1995; Blatter and Niggli, 1998; Egelman
and Montague, 1999).
MATERIALS AND METHODS
Muscle preparation and solutions
All experiments were carried out according to the guidelines laid down by
the local Animal Care Committee. Interossei muscles from BALB/c mice
(2–3 months) were enzymatically isolated (for solutions see below; solu-
tion A) and stored in Ringer’s solution (B) before the experiments as
previously described by Friedrich et al. (1999). Single fibers were then
transferred into the recording chamber, which contained the unbuffered
recording solution (C). Before the second voltage clamp recording (see
below), fibers were perfused up to 20 min with the buffered solution (D)
to allow a complete solution exchange in the TTS. The fibers had lengths
between 490 ?m and 680 ?m and diameters in the range of 35–70 ?m.
For the solutions, concentrations are given in mM: enzymatic isolation
solution (A): 140 NaCl; 4 KCl; 2 CaCl2; 1 MgCl2; 10 HEPES; 1.5 mg/ml
collagenase IA (Sigma Chemicals, St. Louis, MO). Ringer’s solution (B):
140 NaCl; 4 KCl; 2 CaCl2; 1 MgCl2; 10 HEPES; 11 glucose. Unbuffered
isotonic recording solution (C): 10 calcium acetate; 146 TEA-Br; 1 MgCl2;
5 Cs-Br; 5 4-aminopyridine (4-AP); 5 3,4-di-aminopyridine (3,4-DAP); 0.1
KCl; 10 HEPES. Osmolarity was 340 mosmol l?1(Semi-Micro Osmom-
eter Type ML, Knauer, Berlin, Germany). The ionic strength (?/2) was
calculated to be 189 mM. Calcium buffered solution (D): 65 Ca(OH)2; ?5
calcium-acetate; 3.2 Mg(OH)2; 60 TEA-Br; 10 HEPES, 115 malic acid
(Sigma Chemicals); 5 Cs-Br; 5 4-aminopyridine (4-AP); 5 3,4-di-amino-
pyridine (3,4-DAP); 0.1 KCl. The osmolarity was nearly the same as in the
The pH was adjusted to 7.40 ? 0.02 with NaOH or acetate in all
solutions. Voltage-dependent Na?channels were blocked by 500 nM TTX
in (C) and (D).
Two-microelectrode voltage clamp measurements (2-MVC) were carried
out as previously described by Friedrich et al. (1999) using a standard
voltage clamp device (GeneClamp 500, Axon Instruments, Foster City,
CA) with borosilicate pipettes (GB150-F8P) filled with 3 M KCl. Bath
temperature was controlled with an accuracy of ?0.1°C via a custom-built
analog temperature controlling device based on Peltier elements. Most
experiments were carried out at 24°C. In some experiments the temperature
was set to 30°C in order to simulate more closely the physiologically
relevant temperature for mammalian skeletal muscle. Current and voltage
recordings were filtered through an 8-pole Bessel filter and recorded
digitally using pClamp6 software (Axon Instruments) on a standard 486
computer. The sampling rate was set to 400 Hz.
In 10 mM Ca2?solution linear currents during a voltage step pulse were
subtracted by a leak subtraction box, which subtracts a scaled proportion of
the rectangular step pulse from the uncorrected currents recorded (Almers
et al., 1981). When blocking the voltage-dependent Ca2?channels with 2
mM verapamil, depolarization to more positive membrane potentials re-
sulted in nonlinear outward currents (Fig. 1 A), which could not be com-
pensated for by the linear leak subtraction procedure. Similar outward
current components have also been found during the activation of ICain
mammalian extensor digitorum longus fibers (e.g., by Lamb and Walsh,
1987, see their Fig. 1). As these outward currents displayed a distinct
threshold varying from ?5 mV to ?20 mV in the individual fibers (Fig. 1,
B and C; see also Fig. 4) we restricted the membrane potential to quanti-
tatively analyze calcium currents to depolarizations up to maximally
Thus, under our experimental conditions in almost Cl?-free solution
that contained additional Na?- and K?-channel blockers (TTX and TEA,
2 mM verapamil added. (A) Membrane currents are shown in the potential
range between ?100 mV and ?60 mV in unbuffered 10 mM Ca2?
containing hypertonic solution with 2 mM verapamil added to block L-type
Ca2?channels. For potentials more positive than ?20 mV a nonlinear
outward conductance can be seen. (B) The corresponding steady-state I-V
relation between ?40 mV and ?20 mV can be considered linear (r ?
0.996) for the fiber shown in (A), and thus can be removed from the
nonlinear currents by the linear leak subtraction method, so that no addi-
tional outward component interferes with ICain experiments where the
L-type channels are not blocked. (C) Verifies this I-V deviation from the
linear behavior for potentials more positive than ??20 mV in n ? 5 fibers.
Membrane currents in 10 mM Ca2?containing solution with
Ca2?Depletion in Skeletal Muscle2047
Biophysical Journal 80(5) 2046–2055
respectively), mainly ion flow through the L-type DHP Ca2?channel
contributes to the measured inward current signal and the inward currents
shown in the following are almost exclusively slow calcium currents (ICa).
To prevent damage to the fiber due to more vigorous contraction and
shortening at higher membrane potentials, pulse durations were reduced by
100 ms for each additional 10-mV step in membrane potential depolariza-
tion in experiments covering a wider potential range.
For the simulations of the influence of the Ca2?ion con-
centration changes in the TTS on the time course of ICa, the
experimentally recorded current Iexp
buffered solution was used as input data for the numerical
simulation. The influence of ion depletion on the time
course of the Ca2?current when depletion is not suppressed
by a buffering agent was then calculated as Isim
unbuffered solution Iexp
made (similar to Almers et al., 1981): 1) in Ca2?buffered
solution ion depletion in the TTS is completely suppressed.
The currents Iexp
the L-type channel kinetics, which are assumed not to be
influenced by a possible change of the extracellular Ca2?
concentration due to depletion of ions from the lumen of the
TTS. 2) L-type Ca2?channels are exclusively located in the
TTS in differentiated mammalian muscle, therefore the
Ca2?currents only flow through the TTS membrane (see
Fig. 2 B) and not through the surface membrane. 3) In a first
approximation changes in myoplasmic free Ca2?are
We used a modification of a model first proposed by
Adrian et al. (1969a) and by Barry and Adrian (1973) (see
Fig. 2) where the fiber is approximated by a cylinder with
radial symmetry and the complete TTS is simplified as a
homogenous disk with an equivalent radial conductance GL
and an equivalent voltage-dependent wall conductance G?W.
The TTS is connected to the extracellular space via an
access resistance rS. This disk was divided into m concen-
tric shells and the concentration change of Ca2?ions in
each shell was computed numerically in discrete time inter-
buf(t) obtained in calcium-
buf(t). To allow a comparison of the simulation output
unbuf(t) with measurements obtained from the same fiber in
unbuf(t), three main assumptions were
buf(t) measured under these conditions include
The concentration change in the TTS is given by the full
equation of continuity (Barry and Adrian, 1973):
?r??4? ? a ? ICa
? ? zF
4? ? a?Vo? V?r, t? ?RT
denotes the calcium current flow across the tubular wall of
ring n which resembles In
the respective Ca2?concentration (either free or total), cois
the extracellular free Ca2?concentration, R denotes the gas
constant, T is the temperature in K, and F is the Faraday
constant; r is the radial coordinate; t is the time; z ? 2 is the
ion valency; Vois the potential of the sarcoplasm with
respect to the external solution; V(r, t) is the potential at
radial distance r in the lumen of the TTS with respect to the
external solution at time t; a is the fiber radius; ? denotes the
fraction of TTS volume compared to the total fiber volume.
The effective radial diffusion constant for Ca2?ions in
the TTS is given by Adrian et al. (1969a)
trans(see also Appendix). Here, c is
The network-specific factor ? considers the contribution of
the total Ca2?diffusion in radial direction and is described
in detail by Adrian et al. (1969a) for different types of
meshes. Due to the small TTS diameter the Ca2?diffusion
constant in the TTS lumen, DCa
pared to the diffusion constant DCain free solution (Nitsche
and Balgi, 1994):
TTS, is slightly reduced com-
DCa?1 ? 1.125? ln ? ? 1.539? ? 1.2?2
?1 ? ??2
The time dependence of the wall conductance of the TTS,
G?W(t), which forms part of ICa(see Eq. 2), is given by Barry
and Adrian (1973)
G?W?t? ? 2??a?
4 ? G?L
? rS??, (5)
where G?Lis the effective radial conductance (see Adrian et
al. 1969a) and rSis the access resistance as proposed by
Peachey and Adrian (1973). The time-dependent total con-
ductance of the TTS
GT?t? ? fT/Rm?t?
can be calculated from the fraction fTof the current carried
by Ca2?ions compared to the total ionic current flow (Barry
and Adrian, 1973). As L-type Ca2?channels are almost
exclusively located in the TTS (Almers et al., 1981) a
typical value for fTin our simulation was fT? 0.999.
In our approach, we obtained the total time-dependent
membrane resistance per unit area Rm(t) by using the fol-
lowing relationship (Adrian et al., 1970):
Rm?t? ?Vclamp? ECa
2048Friedrich et al.
Biophysical Journal 80(5) 2046–2055
In this equation, where ECadenotes the Nernst potential for
calcium ions, we introduced the time dependence of the
Ca2?current by processing the experimentally recorded
a depolarizing membrane potential Vclampfor a fiber bathed
in Ca2?buffered isotonic solution. In buffered solution,
concentration changes of tubular calcium are not expected.
The predominant mechanism for the decay of ICatherefore
is the voltage-dependent inactivation. To correctly predict
the time course of the calcium current in unbuffered solu-
tion where both the voltage dependence and the tubular
Ca2?depletion are present, it was necessary to consider the
voltage-dependent change of Rm(t) due to the concentration
changes as implied by the Nernst equation.
According to Fig. 2 B (right), the Ca2?concentration
change in a volume element in ring n, [dcn(t)]/dt, is given by
the concentration decrease due to Ca2?efflux from the TTS
into the fiber (In
Ca2?diffusion from (In?1
elements (see Appendix).
buff(t) obtained under voltage clamp conditions with
trans) and the concentration change due to
radial) or into (In
radial) adjacent volume
Voltage dependence of Ca2?conductance
We used a voltage dependence for g ˆ(Vclamp? ECa) as
suggested by Dietze et al. (1998) in mouse myotubes. In
essence, the current-voltage relation is described by a volt-
age-gated Ca2?current with maximal conductance gmax, the
driving force V ? Vclamp? ECa, and experimental reversal
I?V? ? g ˆ?V? ? gmax? ?V ? VCa?
The voltage-dependent activation of the Ca2?conductance
g ˆ is described by a Boltzmann function with the parameters
of half-maximal activation V0.5and the steepness k:
g ˆ?V? ? 1/?1 ? exp??V0.5? V?/k??
The simulation was performed using an ANSI-C program
running on a R10000 workstation (Silicon Graphics, IRIX
6.4). The differential equations were taken from Barry and
Adrian (1973) and integrated using the fourth-order Runge-
Kutta method (Ralston, 1965). The equations for the nu-
merical computation are given in the Appendix.
Stability and convergence of the algorithm used have
been tested and assured with a ?t ? 10?7s for m ? 200
shells or a ?t ? 3 ? 10?5s for m ? 50 shells. The resulting
output data were processed using Origin 5.0 software (Mi-
crocal Software Inc., Northampton, MA).
As a first test of our algorithm we investigated the time
course of the radial Ca2?concentration c(r, t) in the resting
fiber when the external [Ca2?] is suddenly changed from
zero to a new value, c0. c(r, t) is then given with the
diffusion equation in cylindrical coordinates:
ical computations. (A) In a simplified model with cylin-
drical geometry the complete TTS is reduced to a ho-
mogenous disk with equivalent conductances (modified
after Barry and Adrian, 1973). (B) Left: a full schematic
electrical circuit diagram of the equivalent TTS disk
shown in (A) that is used in the numerical simulation.
1/gnindicates the equivalent access resistance of ring n
of the TTS (see Eq. A7 in the Appendix), 1/GLis the
radial resistance as described in the text. Rsmis the
surface membrane resistance excluding the TTS. Gray
arrows indicate the transverse Ca2?current (In
disk n of the TTS into the fiber. (B) Right: part of the
volume element at higher magnification. In
calcium current flow across the tubular wall of ring n
into the fiber. In
radial direction. In?1
ring n ? 1 into ring n of the TTS disk. ?r is the thickness
of a volume element as described in the text.
TTS geometry model used for the numer-
radialis the current flow from ring n in
radialindicates the radial current from
Ca2?Depletion in Skeletal Muscle2049
Biophysical Journal 80(5) 2046–2055
with the boundary condition
c ? c0
for r ? a
at t ? 0
and the initial condition
c ? 0 for 0 ? r ? a
at t ? 0.
The solution of Eq. 10 for the given initial conditions is
? 1 ?2
eff?t?J0?r ? ?n?
J1?a ? ?n?
where J0and J1are Bessel functions of zero- and first-order,
respectively, and the ?nare the positive roots of J0(a ? ?n) ? 0.
The analytical solution of Eq. 11 was calculated with an
algorithm in which the infinite sum is approximated by the
first 100 terms (Uttenweiler et al., 1998) using the Maple V
Rel.3 software (Waterloo Maple Inc., Ontario, Canada).
The diffusion coefficient DCa? 7 ? 10?6cm2/s for Ca2?
in free solution was taken from Cannell and Allen (1984).
Considering the hindered diffusional access to the TTS, Eq.
4 results in a slightly reduced diffusion coefficient DCa
6.83 ? 10?6cm2/s with radiusCa2? ? 0.106 nm (Emsley,
1989) and radiusTTS? 25 nm. For the network considered
here (? ? 0.375, Barry and Adrian, 1973) Eq. 3 leads to a
resulting effective diffusion coefficient for Ca2?ions in the
lumen of the transverse tubular system of DCa
2.59 ? 10?6cm2/s.
In Fig. 3, the results of the analytical concentration pro-
files for simple diffusion into the transverse tubular system
(circles) are compared with the numerical output (lines) of
the Runge-Kutta algorithm. The numerical computation de-
viated ?0.1% from the analytical solution (see Eq. 11).
Voltage clamp data as input for the
As described in the Methods section, the experimentally
recorded currents Iexp
solution were taken as inputs for the numerical simulations.
Based on Iexp
course of the currents was calculated when depletion was
not suppressed by a buffering agent in the external solution.
In Fig. 4, the influence of Ca2?depletion on ICarecordings
obtained from two representative single fibers is shown at
membrane potentials of ?20 mV, 0 mV, and ?20 mV (Fig.
4 A) and at membrane potentials of ?20 mV, ?10 mV, 0
mV, ?10 mV, ?20 mV, and ?30 mV (Fig. 4 B). Fiber
dimensions were: diameter 2a ? 48 ?m, length l ? 640 ?m
for the fiber shown in (A), and 2a ? 56 ?m and length l ?
592 ?m for the fiber shown in (B). From these fibers the
recorded ICameasurements in buffered solution Iexp
red line) and in unbuffered solution Iexp
line) are shown in each panel. Iexp
buf(t) obtained in isotonic buffered Ca2?
buf(t) the influence of ion depletion on the time
unbuf(t) (solid black
unbuf(t) is then compared
with the calculated numerical output Isim
tion model for the unbuffered trace (blue open circles) for a
fixed parameter combination of access resistance rsand
fraction of volume ? occupied by the TTS. For the two
fibers shown the parameters rs? 50 ?cm2, ? ? 0.0052
(Fig. 4 A) and rs? 125 ?cm2, ? ? 0.0053 (Fig. 4 B) were
kept constant for all membrane potentials during the simu-
lation. As can be seen, for potentials up to 0 mV Isim
predicted very well the time course of ICain unbuffered
solution, i.e., Iexp
fibers still showed good agreement between Iexp
the additional outward component described in Fig. 1 be-
came more prominent, as shown in the fiber of Fig. 4 B. For
seven fibers tested at a temperature of 24°C the mean
parameters of rsand ? for which the model most accurately
reproduced the experimental ICatraces for a fixed parameter
combination for all potentials tested were rs? 97.7 ? 44.2
?cm2and ? ? 0.0048 ? 0.0006. The time constants for the
decay of ICain unbuffered solution ?uwere obtained from
the output of the numerical simulation of the experimentally
recorded ICadata in buffered solution. For the representative
fibers shown in Fig. 4, A and B these time constants were
(A): ?u? 40 ms at 0 mV and ?u? 72 ms at ?20 mV;
whereas in (B): ?u? 167 ms at ?10 mV, ?u? 80 ms at 0
mV, ?u? 98 ms at ?10 mV, ?u? 113 ms at ?20 mV, and
?u? 129 ms at ?30 mV. The ICadecay time constants for
the experimental traces in buffered solution were ?b? 254
unbuf(t) of the simula-
unbuf(t). For more positive potentials some
unbuf(t) as the one shown in Fig. 4 A, whereas in other fibers
Ca2?distribution in the TTS of a murine muscle fiber. Numerical simu-
lation (lines) and analytical solution (circles) of the Ca2?distribution for
inward Ca2?diffusion into the TTS in the absence of any applied voltage
and zero initial [Ca2?] within the TTS. Fiber radius is a ? 50 ?m. Initial
conditions are described in the text.
Numerical versus analytical solutions of diffusion-dependent
2050Friedrich et al.
Biophysical Journal 80(5) 2046–2055