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: email@example.com.
© 2001 by the Biophysical Society
2046Biophysical JournalVolume 80 May 2001 2046–2055
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 Muscle 2047
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 Muscle 2049
Biophysical Journal 80(5) 2046–2055
with the boundary condition
c ? c0
for r ? a
at t ? 0
and the initial condition
c ? 0for 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
ms at 0 mV, ?b? 312 ms at ?20 mV (A), and ?b? 255 ms
at ?10 mV, ?b? 118 ms at 0 mV, ?b? 184 ms at ?10 mV,
?b? 235 ms at ?20 mV, and ?b? 184 ms at ?30 mV (B).
In Fig. 5 the averaged ICadecay time constants for different
fibers bathed in isotonic calcium unbuffered (?u, solid bars)
and calcium buffered solution (?b, dashed bars) at three
different membrane potentials (?10 mV, 0 mV, and ?10
mV) are shown. The mean time constants in the buffered
solution (?b) were 230 ? 75 ms at ?10 mV (n ? 4), 144 ?
39 ms at 0 mV (n ? 7), and 144 ? 31 ms at ?10 mV (n ?
experimentally recorded ICain buffered isotonic solution Iexp
two representative fibers bathed in isotonic solution at 30°C for three potentials (?20 mV, 0 mV, and ?20 mV, A) and at 24°C for six membrane potentials
(?20 mV, ?10 mV, 0 mV, ?10 mV, ?20 mV, and ?30 mV, B). Fiber dimensions were: radius a ? 28 ?m, length l ? 592 ?m (A) and a ? 24 ?m,
l ? 640 ?m (B). A constant parameter combination for the access resistance rsand the fraction of volume occupied by the TTS ? was used for all potentials
in the simulation: (A) rs? 50 ?cm2, ? ? 0.0052, and rs? 125 ?cm2, ? ? 0.0053 (B). Iexp
the simulated time course of ICain unbuffered solution Isim
the experimental ICain unbuffered solution declines faster for higher depolarizations (e.g., more positive than 0 mV, B) than predicted by the model
presumably due to an additional outward component as shown in Fig. 1.
The experimental time course of the decay of ICain unbuffered solution is well-predicted by the results of the model. Time courses of the
buf(t) (solid red line) and in unbuffered solution Iexp
unbuf(t) (solid black line) versus simulation for
buf(t) was used as input data for the numerical simulation and
unbuf(t) is shown (blue circles) for each membrane potential. Note that the experimental time course
unbuf(t) is well-reproduced by the numerical simulation for membrane potentials up to ?20 mV in (A) and up to 0 mV in (B). In some fibers (e.g., B)
Ca2?Depletion in Skeletal Muscle2051
Biophysical Journal 80(5) 2046–2055
6). The ?bvalues were significantly larger than the time
constants in unbuffered solution ?u(p ? 0.013 for all
potentials tested, Student’s unpaired t-test) which were
92 ? 36 ms at ?10 mV (n ? 8), 75 ? 20 ms at 0 mV (n ?
9), and 80 ? 20 ms at ?10 ms (n ? 9). On average, the
ratios ?b/?uwere 2.51 at ?10 mV, 1.92 at 0 mV, and 1.79
at ?10 mV.
Resulting calcium concentration in the TTS
As an example for the size of the tubular Ca2?depletion the
time course of the radial calcium concentration [Ca2?](r, t)
for the fiber shown in Fig. 4 A is shown in Fig. 6. The left
plot shows the time- and spatial-dependent [Ca2?](r, t) for a
membrane potential of 0 mV, the right plot for a membrane
potential of ?20 mV. The steady-state averaged [Ca2?]
values for this fiber were 7.5 mM at ?20 mV (not shown),
1.05 mM at 0 mV, and 2.56 mM at ?20 mV; whereas the
calculated steady-state [Ca2?] values in the fiber center
were 6.5 mM at ?20 mV (not shown), 0.7 mM at 0 mV, and
1.53 mM at ?20 mV. Note that the calcium concentration in
the fiber center (r/a ? 0) drops to a value of ?7% of its
The aim of this work was to study whether the faster decline
of slowly activating L-type calcium currents ICain single
mammalian muscle fibers under maintained depolarization
might be explained by the substantial change in ion con-
centration in the small lumen of the tubular system. In
skeletal muscle fibers the decline of Ca2?currents in phys-
iologically isotonic unbuffered Ca2?solution is up to three
times faster than currents recorded from the same fiber in
Ca2?buffered solution, where no Ca2?depletion should
occur (Friedrich et al., 1999).
It should be noted that despite the very detailed and
elegant investigations of the electrical properties of the TTS
(Gage and Eisenberg, 1969; Eisenberg and Gage, 1969;
buffered (?b) and unbuffered (?u) solution obtained from the numerical
simulation. Comparison of the decay time constants at membrane poten-
tials of ?10 mV, 0 mV, and ?10 mV in unbuffered (?u, dashed bars) and
buffered (?b, solid bars) solution. Data are given as mean ? SD. The time
constants had significant smaller values in unbuffered solution compared to
buffered solution for all potentials tested (p ? 0.0012). On average, the
ratios ?b/?uwere 2.51 at ?10 mV, 1.92 at 0 mV, and 1.79 at ?10 mV.
ICadecay time constants for single fibers bathed in calcium-
time-dependent calcium concentration [Ca2?](r, t) in the transverse tubular system for the representative single fiber shown in Fig. 4 A at a membrane
potential of 0 mV (left plot) and ?20 mV (right plot). Note that the calcium concentration in the fiber center (r/a ? 0) drops to a value of ?7% of its initial
value. The steady-state averaged [Ca2?] values for this fiber were 1.05 mM at 0 mV and 2.56 mM at ?20 mV, whereas the calculated steady-state [Ca2?]
values in the fiber center were 0.7 mM at 0 mV and 1.53 mM at ?20 mV.
Radial and time-dependent calcium concentration profiles in a single fiber at different membrane potentials. The resulting radial and
2052Friedrich et al.
Biophysical Journal 80(5) 2046–2055
Valdiosera et al., 1974a, 1974b; Eisenberg et al., 1977;
Mathias et al., 1977; Levis et al., 1983) there is still a lack
of quantitative knowledge about the time- and space-depen-
dent ion concentration changes caused by membrane cur-
rents. There are also very detailed structural investigations
of the TTS (for review see, e.g., Peachey and Franzini-
Armstrong, 1983) and different models for its mathematical
treatment (for review see, e.g., Eisenberg, 1983). However,
as we focused on the ionic current and the concentration
changes, we decided to carry out simulations based on the
early work of Adrian et al. (1969a, 1969b).
The analysis was simplified by the fact that the Ca2?
current flows almost exclusively across the TTS membrane.
Therefore it is possible to use the Ca2?current measure-
ment in Ca2?buffered solution as an input for the calcula-
tion of the time course of depletion development in the TTS.
To introduce the voltage dependence of ICaactivation, we
basically used data by Dietze et al. (1998), as they have
been obtained in skeletal myotubes, though our data can
also be fitted by the voltage dependence introduced by Jafri
et al. (1998) for cardiac tissue.
Aside from the fixed parameters listed in Table 1 we had
to adjust only the values of ? and rsto accurately fit our
experimental data under experimental conditions where the
outward component could be neglected. For the fractional
lumen of the TTS we found a mean value of ? ? 0.0048 ?
0.0006, which agrees well with that of between 0.005 and
0.006 determined in mammalian laryngeal and sternomas-
toid muscle fibers by Hinrichsen and Dulhunty (1982).
Besides the lumen of the TTS, we found the access resis-
tance rsto be the essential parameter for adjusting the
steady-state value of the simulation output Isim
experimentally recorded current Iexp
fibers in isotonic solution was obtained with values ranging
from rs? 35 ?cm2to rs? 175 ?cm2. Our range with a
mean value of rs? 97.7 ? 44.2 ?cm2is in good agreement
with that found in other studies. A value of rs? 150 ?cm2
was proposed by Adrian and Peachey (1973). In a previous
study on the potential distribution in the TTS after applica-
tion of supercharging pulses, Kim and Vergara (1998) re-
ported values in the range of 110–150 ?cm2for frog
unbuf(t) to the
unbuf(t). The best fit for
skeletal muscle. From impedance measurements of frog
skeletal muscle, Valdiosera et al. (1974b) found access
resistance values in the range of 120–130 ?cm2. The lower
part of our range is consistent with values of 20–50 ?cm2
reported by Heiny et al. (1983) obtained from a fit to
absorbance of a potentiometric dye transient. For rat skeletal
muscle Simon and Beam (1985) report a value of rs? 60
?cm2obtained from a fit to charge movements.
With our approach, we are able to directly calculate and
quantify the influence of ion depletion on the time course of
the slowly activating calcium current for a skeletal muscle
fiber. For a representative fiber with a diameter of 48 ?m
and a fiber length of 640 ?m, we found that the Ca2?
concentration in the fiber center can drop to a value of only
7% of the initial value. In some fibers with a larger diameter
we observed a reduction to a minimal value of 3% of the
initial Ca2?concentration (not shown). Such pronounced
Ca2?depletion is also likely to occur in other cells with
diffusion-restricted spaces, e.g., in sheep Purkinje strands
(Levis et al., 1983) and in neural tissue (Egelman and
Montague, 1999). Furthermore, our simulated data in un-
buffered solution Isim
experimental data Iexp
Ca2?-induced ICainactivation as proposed by Neely et al.
(1994) for the cardiac Ca2?channel alpha 1 subunit (alpha
1C) expressed in Xenopus oocytes.
In the present study we confirm the hypothesis that Ca2?
depletion in the transverse tubular system of single skeletal
muscle fibers significantly influences the time course of
ionic currents. Interestingly, our results are in agreement
with previous findings on the time course of ICain cardiac
sheep Purkinje strands (Levis et al., 1983) where the deple-
tion of calcium was enough to explain the turn-off of
calcium current without invoking the inactivation process.
Furthermore, Blatter and Niggli (1998) used laser-scan-
ning confocal microscopy to detect extracellular [Ca2?] and
changes of t-tubular [Ca2?] in cardiac myocytes. The au-
thors reported a slow exchange of solution in the t-tubular
system, suggesting a depletion of Ca2?ions in the TTS of
cardiac myocytes during physiological activity. Thus, it
would be most interesting to test this suggestion by follow-
unbuffis in very good agreement with the
unbuffwithout the need of introducing a
TABLE 1Parameters used for simulations
Definition Symbol and ValueReference
Free Ca2?diffusion constant
Ca2?diffusion constant in tubular lumen
Effective radial diffusion coefficient in the TTS
Ca2?equilibrium potential (23°C)
Conductance of tubular lumen
Fraction of Ca2?conductance in the TTS of total conductance
Number of rings
Size of time step
Co? 10 mM
Ci? 100 nM
ECa? ?144 mV
? ? 0.375
GL? 0.01 S/cm
m ? 200
?t ? 2.5 ? 10?8s
free? 7 ? 10?6cm2/s
TTS? 6.9 ? 10?6cm2/s
eff? 2.59 ? 10?6cm2/s
Cannell and Allen, 1984
Nitsche and Balgi, 1994
Adrian et al., 1969a
Barry and Adrian, 1973
Ca2?Depletion in Skeletal Muscle 2053
Biophysical Journal 80(5) 2046–2055
ing the changes in tubular Ca2?concentration in mamma-
lian muscle with fluorescence indicators and apply our
combined experimental and simulation approach.
The aim of this appendix is to show our extension of the model developed
by Barry and Adrian (1973). The simulations were based on the original
source code, developed by P. H. Barry for his simulation of potassium
depletion in the TTS of skeletal muscle fibers during hyperpolarizing
pulses. Here we present the equations used in the numerical model, all of
which are taken from Barry and Adrian (1973). The basic approach for the
model was given by:
dt? f?t, cn?
cn?t ? ?t? ? cn?t? ??K1n? 2K2n? 2K3n? K4n?
where cndenominates the respective Ca2?concentration in the nth ring, t
is the time, Kinare the Runge-Kutta coefficients specified below, and
f(t, cn) is given by the difference between the loss of Ca2?due to the
transport number effect (caused by the current through the TTS wall) and
the increase due to radial diffusion, as shown in Eqs. 1 and 2.
f?t, cn? ? a1n? ?cn?1?t? ? cn?t?? ? a2n? ?cn?t? ? cn?1?t??
eff?m ? n ? 1?/???r?2? ?m ? n ? 0.5??
eff?m ? n?/???r?2? ?m ? n ? 0.5??
a3n? a/?? ? F ? ??r?2? ?m ? n ? 0.5??,
1), a the fiber radius, n the actual and m the total number of concentric
shells of the TTS disk, F the Faraday constant, ? the fraction of the fiber
volume occupied by the TTS, r the radius of the concentric shell, and ?r
the thickness of a single ring given by ?r ? a/m.
At the fiber surface (n ? 1) A3 is given by
effas the effective radial diffusion coefficient in the TTS (see Table
f?t, c1? ? a1n? ?c0?t? ? ?c1?t? ? 0.5 ? ?c1?t? ? c2?t????
?a21? ?c1?t? ? c2?t?? ? a31? I1
a11? ?m ? DCa
eff?/??r ? rS? G?L? ?m ? 0.5??,
with the access resistance rsof the tubular lumen and G?Lthe effective
radial conductance (Adrian et al., 1969a).
The Runge-Kutta coefficients are
K1n? ?t ? f?t, cn?
K2n? ?t ? f?t ??t
K3n? ?t ? f?t ??t
K4n? ?t ? f?t ? ?t, cn? K3n?.(A4d)
The current In
transof ring n passing through the TTS wall into the fiber is
trans?t? ? Gn?t? ??m ? n ? 0.5?
rel?t? ? V?t? ? ECa? Vo? Vn?t? ?
2 ? Fln?
is the driving force for the ionic current and
Gn?t? ? G?W?t? ? g ˆ?V? ? ??r?2?m ? n ? 0.5?/a
is the conductance of ring n. G?Wis the voltage-dependent wall conductance
of the TTS, g ˆ(V) is the apparent driving force for ICa, and Vois the
membrane potential during the voltage clamp pulse. Vnis the potential
difference at the center of ring n with respect to the external solution and
is given by
radial?t? ? a ? ln?
Vn?1?t? ? Vn?t? ?
m ? n ? 0.5
m ? n ? 0.5?
where GLis an equivalent radial conductance of the simplified homoge-
nous disk of the TTS. In
toward the fiber center and is given by:
radial(t) represents the radial current leaving ring n
radial?t? ? In?1
radial?t? ? In
The termination criteria for the calculation of ?I ? Im?1
to ?I ? 10?20.
This work was supported by a grant from the German Research Foundation
Deutsche Forschungsgemeinschaft (DFG, Graduiertenkolleg 388, “Bio-
technologie”) and DFG research unit Image Sequence Analysis to Inves-
tigate Dynamic Processes, FOR 240/2-2. This work is part of a Ph.D. thesis
of T. Ehmer.
Adrian, R. H., W. K. Chandler, and A. L. Hodgkin. 1969a. The kinetics of
mechanical activation in frog muscle. J. Physiol. 204:207–230.
Adrian, R. H., W. K. Chandler, and A. L. Hodgkin. 1970. Voltage clamp
experiments in striated muscle fibres. J. Physiol. 208:607.
Adrian, R. H., L. L. Costantin, and L. D. Peachey. 1969b. Radial spread of
contraction in frog muscle fibres. J. Physiol. 204:231–257.
Adrian, R. H., and L. D. Peachey. 1973. Reconstruction of the action
potential of frog sartorius muscle. J. Physiol. 235:103–131.
Almers, W., R. Fink, and P. T. Palade. 1981. Calcium depletion in frog
muscle tubules: the decline of calcium current under maintained depo-
larisation. J. Physiol. 312:177–207.
Almers, W., and P. T. Palade. 1981. Slow calcium and potassium currents
across frog muscle membrane: measurements with a vaseline-gap tech-
nique. J. Physiol. 312:159–176.
Amsellem, J., R. Delorme, C. Souchier, and C. Ojeda. 1995. Transverse-
axial tubular system in guinea pig ventricular cardiomyocyte: 3D recon-
struction, quantification and its possible role in K?accumulation-
depletion phenomenon in single cells. Biologie Cellulaire 85:43–54.
2054Friedrich et al.
Biophysical Journal 80(5) 2046–2055
Barry, P. H., and R. H. Adrian. 1973. Slow conductance changes due to Download full-text
potassium depletion in the transverse tubules of frog muscle fibers
during hyperpolarizing pulses. J. Membr. Biol. 14:243–292.
Beaty, G. N., and E. Stefani. 1976. Inward calcium current in twitch muscle
fibres of the frog. J. Physiol. 260:27P.
Blatter, L. A., and E. Niggli. 1998. Confocal near membrane detection of
calcium in cardiac myocytes. Cell Calcium. 23:269–279.
Cannell, M. B., and D. G. Allen. 1984. Model of calcium movement during
activation in the sarcomere of frog skeletal muscle. Biophys. J. 45:
Cota, G., L. Nicola-Siri, and E. Stefani. 1983. Calcium-channel gating in
frog skeletal muscle membrane: effects of temperature. J. Physiol.
Cota, G., L. Nicola-Siri, and E. Stefani. 1984. Calcium channel inactivation
in frog (Rana pipiens and Rana montezuma) skeletal muscle fibres.
J. Physiol. 354:99–108.
Cota, G., and E. Stefani. 1989. Voltage-dependent inactivation of slow
calcium channels in intact twitch muscle fibers of the frog. J. Gen.
Crank, J. 1975. The Mathematics of Diffusion. Oxford University Press.
Dietze, B., F. Bertocchini, V. Barone, A. Struck, V. Sorrentino, and W.
Meltzer. 1998. Voltage-controlled Ca2?release in normal and ryanodine
receptor type 3 (RyR3)-deficient mouse myotubes. J. Physiol. 513.1:
Dulhunty, A. 1982. Effect of chloride withdrawal on the geometry of the
T-tubules in amphibian and mammalian muscle. J. Membr. Biol. 67:
Egelman, D. M., and P. R. Montague. 1999. Calcium dynamics in the
extracellular space of mammalian neural tissue. Biophys. J. 76:
Eisenberg, R. S. 1983. Impedance measurement of the electrical structure
of skeletal muscle. In Handbook of Physiology, Section 10: Skeletal
Muscle, Chapter 2. L. D. Peachey, R. H. Adrian, and S. R. Geiger,
editors. American Physiological Society, Bethesda, MD. 301–324.
Eisenberg, R. S., and P. W. Gage. 1969. Ionic conductances of the surface
and transverse tubular membranes of frog sartorius fibers. J. Gen.
Eisenberg, R. S., R. T. Mathias, and J. S. Rae. 1977. Measurement,
modeling, and analysis of the linear electrical properties of cells. Ann.
NY. Acad. Sci. 303:342–354.
Emsley, J. 1989. The Elements. Clarendon Press, London.
Francini, F., L. Pizza, and G. Traina. 1992. Inactivation of the slow calcium
currents in twitch skeletal muscle fibres of the frog. J. Physiol. 448:
Francini, F., and E. Stefani. 1989. Decay of the slow calcium current in
twitch muscle fibers of the frog is influenced by intracellular EGTA.
J. Gen. Physiol. 94:953–969.
Friedrich, O., T. Ehmer, and R. H. A. Fink. 1999. Calcium currents during
contraction and shortening in enzymatically isolated murine skeletal
muscle fibres. J. Physiol. 517.3:757–770.
Gage, P., and R. S. Eisenberg. 1969. Capacitance of the surface and
transverse tubular membrane of frog sartorius muscle fibers. J. Gen.
Garcı ´a, J., K. McKinley, S. H. Appel, and E. Stefani. 1992. Ca2?current
and charge movement in adult single human skeletal muscle fibres.
J. Physiol. 454:183–196.
Heiny, J. A., F. M. Ashcroft, and J. M. Vergara. 1983. T-system optical
signals associated with inward rectification in skeletal muscle. Nature.
Hinrichsen, C., and A. Dulhunty. 1982. The contractile properties, histo-
chemistry, ultrastructure and electrophysiology of the cricothyroid and
posterior cricoarytenoid muscles in the rat. J. Muscle Res. Cell Motil.
Jafri, M. S., J. J. Rice, and R. L. Winslow. 1998. Cardiac Ca2?dynamics:
the roles of ryanodine receptor adaption and sarcoplasmic reticulum
load. Biophys. J. 74:1149–1169.
Kim, A. M., and J. L. Vergara. 1998. Supercharging accelerates T-tubule
membrane potential changes in voltage clamped frog skeletal muscle
fibers. Biophys. J. 75:2098–2116.
Lamb, G. D., and T. Walsh. 1987. Calcium currents, charge movement and
dihydropyridine binding in fast- and slow-twitch muscles of rat and
rabbit. J. Physiol. 393:595–617.
Levis, R. A., R. T. Mathias, and R. S. Eisenberg. 1983. Electrical properties
of sheep Purkinje strands. Electrical and chemical potentials in the clefts.
Biophys. J. 44:225–248.
Lorkovic, H., and R. Ru ¨del. 1983. Influence of divalent cations on potas-
sium contracture duration in frog muscle fibres. Pflu ¨gers Arch. 398:
Mathias, R. T., R. S. Eisenberg, and R. Valdiosera. 1977. Electrical
properties of frog skeletal muscle fibers interpreted with a mesh model
of the tubular system. Biophys. J. 17:57–93.
Miledi, R., I. Parker, and P. H. Zhu. 1983. Changes in threshold for calcium
transients in frog skeletal muscle fibres owing to calcium depletion in
the T-tubules. J. Physiol. 344:233–241.
Neely, A., R. Olcese, X. Wei, L. Birnbaumer, and E. Stefani. 1994.
Ca2?-dependent inactivation of a cloned cardiac Ca2?channel alpha 1
subunit (alpha 1C) expressed in Xenopus oocytes. Biophys. J. 66:
Nicola-Siri, L., J. A. Sanchez, and E. Stefani. 1980. Effect of glycerol
treatment on the calcium current of frog skeletal muscle. J. Physiol.
Nitsche, J. M., and G. Balgi. 1994. Hindered Brownian diffusion of
spherical solutes within circular cylindrical pores. Industrial and Engi-
neering Chemistry Research. 33:2242–2247.
Peachey, L. D. 1965. The sarcoplasmic reticulum and transverse tubules of
the frog’s sartorius. J. Cell Biol. 25:209–231.
Peachey, L. D., and R. H. Adrian. 1973. Electrical properties of the
transverse tubular system. In Structure and Function of muscle, Vol. III.
Physiology and Biochemistry. 2nd Ed. G. H. Bourne, editor. Academic
Press, Inc., New York.
Peachey, L. D., and C. Franzini-Armstrong. 1983. Structure and function of
membrane systems of skeletal muscle. In Handbook of Physiology,
Section 10: Skeletal Muscle, Chapter 2. L. D. Peachey, R. H. Adrian and
S. R. Geiger, editors. American Physiological Society, Bethesda, MD.
Ralston, A. 1965. A First Course in Numerical Analysis. McGraw Hill
Book Co., Inc., New York.
Sa ´nchez, J. A., and E. Stefani. 1978. Inward calcium current in twitch
muscle fibres of the frog. J. Physiol. 283:197–209.
Sa ´nchez, J. A., and E. Stefani. 1983. Kinetic properties of calcium channels
of twitch muscle fibres of the frog. J. Physiol. 337:1–17.
Simon, B. J., and K. G. Beam. 1985. The influence of transverse tubular
delays on the kinetics of charge movement in mammalian skeletal
muscle. J. Gen. Physiol. 85:21–42.
Stanfield, P. R. 1977. A calcium dependent inward current in frog skeletal
muscle fibres. Pflu ¨gers Arch. 368:267–270.
Uttenweiler, D., and R. H. A. Fink. 1999. Mathematical modeling of
Ca2?-transients measured with fluorescence imaging. In Handbook of
Computer Vision and Applications. B. Ja ¨hne, H. Haussecker, and P.
Geissler, editors. Academic Press, New York. 737–750.
Uttenweiler, D., C. Weber, and R. H. A. Fink. 1998. Mathematical mod-
eling and fluorescence imaging to study the Ca2?turnover in skinned
muscle fibers. Biophys. J. 74:1640–1653.
Valdiosera, R., C. Clausen, and R. S. Eisenberg. 1974a. Circuit models of
the passive electrical properties of frog skeletal muscle fibers. J. Gen.
Valdiosera, R., C. Clausen, and R. S. Eisenberg. 1974b. Impedance of frog
skeletal muscle fibers in various solutions. J. Gen. Physiol. 63:460–491.
Ca2?Depletion in Skeletal Muscle 2055
Biophysical Journal 80(5) 2046–2055