Electrophysiological Modeling of Fibroblasts and their Interaction with Myocytes

Electrophysiological Modeling of Fibroblasts and their Interaction

with Myocytes

FRANK B. SACHSE, ALONSO P. MORENO, and J. A. ABILDSKOV

Nora Eccles Harrison Cardiovascular Research and Training Institute, University of Utah, 95 S 2000 E, Salt Lake City,UT 84112, USA

(Received 7 December 2006; accepted 2 November 2007; published online 13 November 2007)

Abstract—Experimental studies have shown that cardiacfibroblasts are electrically inexcitable, but can contribute toelectrophysiology of myocardium in various manners. Theaim of this computational study was to give insights in theelectrophysiological role of fibroblasts and their interactionwith myocytes. We developed a mathematical model offibroblasts based on data from whole-cell patch clamp andpolymerase chain reaction (PCR) studies. The fibroblastmodel was applied together with models of ventricularmyocytes to assess effects of heterogeneous intercellularelectrical coupling. We investigated the modulation of actionpotentials of a single myocyte varying the number of coupledfibroblasts and intercellular resistance. Coupling to fibro-blasts had only a minor impact on the myocyte’s resting andpeak transmembrane voltage, but led to significant changesof action potential duration and upstroke velocity. Weexamined the impact of fibroblasts on conduction inone-dimensional strands of myocytes. Coupled fibroblastsreduced conduction and upstroke velocity. We studiedelectrical bridging between ventricular myocytes via fibro-blast insets for various coupling resistors. The simulationsshowed significant conduction delays up to 20.3 ms. Insummary, the simulations support strongly the hypothesisthat coupling of fibroblasts to myocytes modulates electro-physiology of cardiac cells and tissues.

Keywords—Computational modeling, Fibroblast, Cardiac

electrophysiology, Electrical bridging.

INTRODUCTION

Fibroblasts are by number the predominant cardiaccell type and organized in a three-dimensional networkin the myocardium.16 Fibroblasts are the primaryproducer of myocardial extracellular matrix proteins.12

Additionally, they seem to play a role in cardiac elec-trophysiology. Several cardiac diseases are associatedwith activation of fibroblasts and transitions of their

phenotype.2 Myocardial infarction and injury activatefibroblasts as major player of the healing response, butchronic activation can lead to fibrosis, hypertrophy,and heart failure.31 In the aged and hypertrophic heart,fibroblasts produce collagenous septa, which can act asbarriers for electrical conduction.

Cardiac fibroblasts are electrically inexcitable, butcan contribute to the electrophysiology of myocytes invarious manners. Electrical coupling via fibroblastswas hypothesized to underlie synchronization of con-traction among individual myocytes, which was re-ported in early studies of cells cultures from mouse andrat embryos.17,20 Electrical coupling of fibroblasts andmyocytes was demonstrated at cellular and tissue levelas well as in cell cultures.7,14,27,37 Coupling of fibro-blasts to myocytes was reported via Cx43 gap junctionsin sheep ventricular and via Cx45 in rabbit sinoatrialnode cells.5,26 Resistances between myocytes andfibroblasts in the range of 125 and 3225MX have beenmeasured in cell culture of neonatal rat cardiac cells.37

Fibroblast–myocyte coupling via both types of gapjunctions has been demonstrated in sheep ventricularscars.4 Additionally, fibroblast–fibroblast coupling viaCx40 gap junctions and a resistance in the range of 166and 5714MX have been reported in culture of neonatalrat cells.6,37 Resistances in myocyte–fibroblast andfibroblast–fibroblast pairs are significantly smallerthan in myocyte pairs. For instance, a mean resistanceof 5.7 and 5:9MX has been measured in pairs of rabbitventricular and atrial myocytes, respectively.

Several studies suggested electrical bridging ofmyocytes by fibroblasts.7,14,37 Synchronous contrac-tion in myocyte–fibroblast–myocyte triplets associatedwith a summary resistance between 333 and 6666MXhave been found for culture of neonatal rat cardiaccells.37 In a similar preparation, conduction overdistances up to 300 lm along fibroblast inserts withdelays of up to 68 ms and apparent conductionvelocities of 4.6 ± 1.8 mm/s were reported.14

Address correspondence to Frank B. Sachse, Nora Eccles

Harrison Cardiovascular Research and Training Institute, University

of Utah, 95 S 2000 E, Salt Lake City, UT 84112, USA. Electronic

mail: [email protected]

Annals of Biomedical Engineering, Vol. 36, No. 1, January 2008 ( 2007) pp. 41–56

DOI: 10.1007/s10439-007-9405-8

0090-6964/08/0100-0041/0 2007 Biomedical Engineering Society

41

The role of fibroblasts as electrical bridges betweenmyocytes still remains to be clarified.38 A beneficialrole was proposed by reason that fibroblasts canincrease electrical coupling of myocytes, which areseparated by collagenous septa and scars. Contrarily,electrical bridges were found to be arrhythmogenic invarious studies of Purkinje fibers as well as of atrialand ventricular tissue.1,9–11,47,48 These studies werecarried out with preparations with segmental depressedexcitability and showed in particular slowed conduc-tion and reflected reentry.

The aim of this study was to give insights in theelectrophysiological role of fibroblasts and theirinteraction with myocytes. The study was carried outwith mathematical models and computational simula-tions of cellular and multi-cellular electrophysiology.We developed a novel mathematical model of cardiacfibroblasts on basis of data from recent whole-cellpatch clamp and polymerase chain reaction (PCR)studies. The fibroblast model was applied together withmodels of ventricular myocytes to assess electrophysi-ological effects of heterogeneous electrical coupling ofcells. We investigated the modulation of the actionpotentials of single myocytes by connected fibroblastsand electrical bridging between ventricular myocytesvia fibroblasts. We developed protocols to quantifyconduction delays and attempted to determine condi-tions for reflected reentry.

METHODS

Fibroblast Model Development

Our model describes the transmembrane voltageand currents of fibroblasts on basis of electrophysio-logical and PCR studies of isolated ventricular cellsfrom adult male rats.8,43 These studies indicated thattwo K+ currents, an inwardly rectifying current IKir

and a time- and voltage dependent outward currentIShkr, are major determinants of the fibroblast’s volt-age–current relationship. In addition, our model de-scribes an unspecific background current Ib responsiblefor maintaining the resting transmembrane voltage andincludes a stimulus current Istim. The membrane of afibroblast is considered as introduced by Hodgkin–Huxley as an isopotential compartment with themembrane capacity Cm and transmembrane voltageVm, which is a function of transmembrane currents:

@

@tVm ¼

1

CmIKir þ IShkr þ Ib þ Istimð Þ

Electrophysiological studies have shown that theinwardly rectifying current IKir is nearly instantaneous,modulated by the external K+ concentration, and canbe blocked by BaCl2.

8 PCR measurements have

indicated that channels consisting of Kir subunits arethe major carrier of this current. We described thecurrent IKir by re-parameterizing a model for IK1 incardiomyocytes21:

IKir ¼ GKirOKir

ffiffiffiffiffiffiffiffiffiffiffiffi

½Kþoq

ðVm EKÞ

with the conductance GKir and the reversal voltage EK.The channel open probability OKir was defined as:

OKir ¼1

aKir þ expðbKirðVm EKÞF=RTÞ

with the constants aKir and bKir. The reversal voltageEK is given by the Nernst equation for K+:

EK ¼RT

Fln½Kþo½Kþi

with constants as described in Tables 1 and 2. Theconductance GKir was set according to results of wholecell voltage clamp experiments8 (Table 1).

The outward current IShkr of fibroblasts was foundto be time- and voltage dependent. PCR measurementssuggested that this current passes through channelsconsisting of subunits in the Kv 1.x family (Shaker),which shows delayed rectifier and relatively slowinactivation properties. We reconstructed this currentbased on the Goldman–Hodgkin–Katz current equa-tions19 and a Markovian model of delayed rectifier K+

TABLE 1. Constants of fibroblast model.

Parameter Symbol Value

Temperature (K) T 295

Membrane capacity (pF) Cm 4.50

Intracellular K+ concentration (mM) [K+]i 140

Ikir parameter aKir 0.94

Ikir parameter bKir 1.27

Conductance for Ikir (nS) GKir 1.02

Ishkr parameter (s-1) kv0 30

Ishkr parameter zv 1.28

Ishkr parameter (s-1) k-v0 2

Ishkr parameter z-v -1.53

Ishkr parameter (s-1) ko 77

Ishkr parameter (s-1) k-o 18

Membrane permeability to Ishkr (m3/s) PShkr 5.40 Æ 10-15

Maximum conductance of background

current Ib (pS)

Gb 6.9

Reversal voltage of background

current Ib (mV)

Eb 0.00

TABLE 2. Physical constants.

Name Symbol Value

Faraday’s constant (C mol-1) F 9.65 Æ 104

Gas constant (J K-1 mol-1) R 8.31

SACHSE et al.42

currents and its parameterization for shaker channelsin neurons25,49 (Fig. 1):

IShkr ¼ PShkrOShkrVmF

2

RT

½Kþi ½Kþo expðVmF=RTÞ1 expðVmF=RTÞ

with the permeability PShkr and the probability of theopen state OShkr. The Markovian model describes fiveclosed states, C0Shkr . . .C4Shkr; and one open stateOShkr:

@

@tC0Shkr¼4kvC0ShkrþkvC1Shkr

@

@tC1Shkr¼4kvC0Shkrð3kvþkvÞC1Shkrþ2kvC2Shkr

@

@tC2Shkr¼3kvC1Shkrð2kvþ2kvÞC2Shkrþ3kvC3Shkr

@

@tC3Shkr¼2kvC2Shkrðkvþ3kvÞC3Shkrþ4kvC4Shkr

@

@tC4Shkr¼kvC3Shkrðkoþ4kvÞC4ShkrþkoOShkr

@

@tOShkr¼koC4ShkrkoOShkr

The rate coefficients kv and k-v are exponential func-tions of the transmembrane voltage Vm:

kv ¼kv0 exp VmzvF=RTð Þkv ¼kv0 exp VmzvF=RTð Þ

with the parameters kv0 and k-v0 describing associatedrate coefficients at 0 mV, and the apparent amount ofcharge moved between states zv and z-v. The ratecoefficients ko and k-o are constants. The parameterskv0, k-v0, ko0, and k-o0 were adjusted to reconstructexperimental studies of the activation kinetics atvarious [K+]o (Fig. 2). Initial values of the statesC0Shkr - C4Shkr, and OShkr for resting voltage are givenin Table 3.

A description of an unspecific background currentIb was added to the fibroblast model. Backgroundcurrents are usually observed in whole cell patch-clampexperiments, e.g., with rat kidney fibroblasts.18 Thebackground current Ib was described as in previousohmic models of kidney and cardiac fibroblasts28,45:

Ib ¼ GbðVm EbÞ

The reversal potential Eb was set to 0 mV18 and thebackground conductance Gb was adjusted to fix theresting voltage of an isolated fibroblast to -58 mV for[K+]o = 5 mM43 (Fig. 3).

The system of ordinary differential equationsunderlying the fibroblast model was solved with theforward Euler method using a time step of 1 ls.36 Allcalculations were performed in double precision float-ing point arithmetic.

Fibroblast–Myocyte Models

Fibroblast–myocyte interactions were studied bycomputational experiments, in which electrophysio-logical models of rat left ventricular subepicardialmyocytes34 and the upper fibroblast model were elec-trically connected by resistors. The resistors describethe intracellular space and the intercellular couplingvia gap junctions. We configured four experimentalarrangements (Fig. 4):

ShkrShkr C2Shkr C3Shkr C4ShkrC0Shkr

4kv 3kv 2k v

4k−v3k −v2k−v

k v k o

−vk −okOC1

FIGURE 1. Schematic of Markovian model for the time- andvoltage dependent outward current IShkr (modified from25).The model describes probabilities for six different states of anion channel, i.e., five closed states, C0Shkr - C4Shkr, and anopen state OShkr.

0

10

20

30

40

50

60

10 20 30 40 50

Act

ivat

ion

time

cons

tant

[ms]

Vstep [mV]

FIGURE 2. Activation kinetics of time- and voltage depen-dent outward current IShkr for [K+]o = 5 mM. A voltage clampprotocol was applied to characterize activation time con-stants. Fibroblasts were initially clamped at -90 mV. Voltagesteps Vstep caused activation of the current IShkr, which ischaracterized by activation time constants. Crosses (+) indi-cate measured data.43 Simulated data are represented with theclosed line.

TABLE 3. Initial values of fibroblast model for [K+]o = 5 mM.

Parameter Symbol Value

Transmembrane voltage (mV) Vm -58.00

0th Closed Ishkr state C0Shkr 9.11 Æ 10-1

1st Closed Ishkr state C1Shkr 8.57 Æ 10-2

2nd Closed Ishkr state C2Shkr 3.02 Æ 10-3

3rd Closed Ishkr state C3Shkr 4.74 Æ 10-5

4th Closed Ishkr state C4Shkr 2.79 Æ 10-7

Open Ishkr state OShkr 0

Modeling of Fibroblast–Myocyte Interactions 43

a single myocyte connected to a number of fibro-blasts (M–Fn), a chain of myocytes coupled to an array of fibro-

blasts (Mm–Fnxm), two chains of myocytes connected by a chain of

fibroblasts (Mm–Fn–Mm), and two chains of myocytes with ohmic coupling

(Mm–Mm).

We varied in the computational experiments withthese arrangements, the number of fibroblasts n, theresistor between myocytes and fibroblasts Rmf, and ifapplicable in-between fibroblasts Rff, myocytes Rmm,and chains Rcc.

The arrangement M–Fn (Fig. 4a) served to charac-terize and quantify the impact of various couplingmyocyte–fibroblast resistors Rmf and the number ofcoupled fibroblasts n on the resting and action poten-tial of a myocyte. Coupling resistances Rmf of100MX;1GX; and 10GX were chosen as an estimateof high, average and marginal coupling, respectively.Further simulations with a coupling resistance Rmf of10MX were performed to explore a case of extremelyhigh coupling.

The arrangement Mm–Fnxm (Fig. 4b) allowed us tostudy the effect of fibroblasts on conduction velocity inmyocardium. Myocyte–myocyte coupling was recon-structed by a resistor Rmm of 6MX: The myocyte–fibroblast resistor Rmf was set to 10MX;100MX;1GX;and 10GX: The measurement of conduction velocitywas based on the detection of activation times for the8th and 24th myocyte and assuming a myocyte lengthof 120 lm. The time of maximal upstroke velocity ofthe transmembrane voltage determined activation time.

The arrangement Mm–Fn–Mm (Fig. 4c) was used toprovide insights in the role of fibroblasts for electrical

bridging between myocytes. Myocyte–myocyte resis-tors Rmm and myocyte–fibroblast resistors Rmf weresystematically varied in the range of 1100MX: Thefibroblast—fibroblast resistor Rff was set identical tothe myocyte–fibroblast resistor Rmf. This decision re-duced the dimensionality of parameter variation from3 to 2. Maximal conduction delays for each Rmm weredetected with a bisection method for searching maximain monotonic increasing function of.36 The stoppingcriterion for the search was that the difference betweentwo successively determined resistances Rmf wassmaller than 1X:

Similar studies were carried out with arrangementMm–Mm (Fig. 4d). Here, the myocyte–myocyte resis-tors Rmm and chain–chain resistor Rcc were variedsearching for maximal conduction delays.

Conduction in cell strands was modeled by spatialfinite difference discretization of the cable equation3:

a

2

@

@x

1

qðxÞ@Vmðx; tÞ

@x

¼ Iion þ Cm@Vmðx; tÞ

@t

with the fiber radius a (8 lm), specific membranecapacity Cm, resistivity q, summary membrane currentIion and transmembrane voltage Vm. The resistivity wascalculated from the upper resistors Rmm, Rmf, Rff, andRcc. Neumann boundary conditions were assigned atthe ends of strands. The spatial discretization was setsimilar to reported mean cell lengths: in regions withmyocytes to 120 lm and with fibroblasts to 8 lm.

The ordinary differential equations underlying thearrangements were solved using the Euler method witha time step of 1 ls.36 Intercellular currents were up-dated every 1 ls. In all simulations a stimulus fre-quency of 0.5 Hz was chosen. A stimulus current of5 nA was applied to myocyte 0 until a threshold volt-age was reached. The threshold voltage was -50 mVand -30 mV for the M–Fn and other arrangements,respectively. The resulting stimulus duration was keptshorter than the duration of 5 ms used by Panditet al.34 to avoid significant overlap of sodium andstimulus current. Simulation results were analyzedafter the 10th stimulation.

RESULTS

Fibroblast Model

An electrophysiological model of cardiac fibroblastswas developed based on measurements of isolated ratventricular fibroblasts.8,43 Measured and simulatedvoltage–current relationships for [K+]o = 5, 10, 50,and 100 mM are shown in Fig. 5d and 5e, respectively.The voltage–current relationships describe steady-statecurrents for a voltage clamping protocol, where volt-

-90

-80

-70

-60

-50

-40

-30

-20

-10

1 10 100 1000

Vre

st [m

V]

Gb [pS]

FIGURE 3. Effect of conductance Gb of the unspecificbackground current Ib on resting voltage Vrest of an isolatedfibroblast.

SACHSE et al.44

ages steps varied between -110 and 50 mV (Fig. 5a).Exemplary measured and simulated currents aredemonstrated in Fig. 5b and 5c, respectively, for thesevoltage steps and [K+]o = 10 mM.

Major currents of our model were constituted by theinwardly rectifying current IKir (Fig. 6a) and the time-and voltage dependent outward current IShkr (Fig. 6b),which dominated the voltage–current relationships inthe lower and upper range of the transmembranevoltage Vm, respectively. For physiological [K+]o (5 mM) the lower and upper voltage range was definedas Vm< -90 mV and Vm > -40 mV, respectively.The unspecific background current Ib contributed little(relative to IKir and IShkr) to the summary voltage–current relationship (Fig. 6c) in the upper and lowervoltage ranges. The current Ib was significant onlyoutside of these ranges and caused deviations of theresting voltage of fibroblasts (-59 mV in case ofphysiological [K+]o) from the K+ reversal potential.

Single Myocyte Coupled to Fibroblasts

Simulated transmembrane voltages of myocytes andfibroblasts are demonstrated in Fig. 7a and 7b for thecase of high intercellular coupling with a resistance Rmf

of 100MX: The transmembrane voltage of the fibro-blasts followed closely the transmembrane voltage ofthe myocyte. The action potential duration at 50%repolarization (APD50) progressively increased from

14.9 ms to 22.6 ms as the number of fibroblasts n in-creased (Fig. 7f). Similarly, the action potential dura-tion to 90% repolarization (APD90) increased from38.9 ms to 61.3 ms as the number of fibroblasts n in-creased.

Electrical coupling led to intercellular currents Iinter,which flowed during stimulus and action potentialupstroke from the myocyte to the fibroblasts (Fig. 7c,d). The summary current Iinter from and to the myocyteincreased as the number of fibroblasts n increased.Intercellular currents from myocyte to fibroblasts in-creased their transmembrane voltage by charging themembrane capacitor. Subsequently, the membranecapacitor of the fibroblasts was discharged and thecurrent flowed reversed. This current was prominentonly at period of the action potential notch andrepolarization.

We characterized availability of sodium channels asproduct of fast and slow inactivation gating variablesof INa (Fig. 7e). Increasing fibroblast number pro-longed inactivation. The delay after stimulus to reap-pearance of 1% availability was 47.7, 50.3, 60.8, and74.1 ms for 0, 1, 5, and 10 coupled fibroblasts.

In order to explore the role of intercellular couplingin modulation of action potentials, further simulationswere performed with an intercellular resistances Rmf of10MX;1GX, and 10GX (Fig. 8). Similar transmem-brane voltages and APDs as for a coupling resistanceRmf ¼ 100MX were found for a simulation with a

(d)

n

Mm+1 M2m

Fn,mFn,1

Fn

M

F1

F2

M

F

1

1,1

M −Fm nxm

F1,m

Mm

mM −M

m nM −F −M1 Mm Mm+1 M2m mn1 FF

1 m

M

MM m

(a)

(b)

(c)

M−F

FIGURE 4. Arrangements of myocytes and fibroblasts for computational studies.


resistance Rmf ¼ 10MX (Fig. 8a–c). In case of resis-tances Rmf 1GX the transmembrane voltage offibroblasts did not resemble the voltage of the myocyte(Fig. 8e, h). Marginal electrical coupling,Rmf ¼ 10GX; led to negative n-APD relationships(Fig. 8i). The APD50 and APD90 decreased from 14.9to 13.6 ms and from 39.0 to 37.1 ms, respectively, asthe number of fibroblasts n increased.

Resting transmembrane voltages and maximal up-stroke velocities are presented in Fig. 9 dependent onthe number of coupled fibroblasts. The resting voltageof a myocyte without coupled fibroblasts was -81.1 mV (Fig. 9a). The membrane voltage wasincreasingly depolarized with increasing number ofcoupled fibroblasts, though the changes were marginal.The most depolarized membrane voltage correspondedto a resistor Rmf ¼ 10MX and coupling to 10 fibro-blasts, which caused depolarization to -80.6 mV. Themaximal upstroke velocity dV/dtmax of an uncoupledmyocyte was 160 V/s (Fig. 9b). With 10 coupledfibroblasts, the velocity decreased to 114, 130, 150, and

157 V/s in case of Rmf ¼ 10MX;100MX;1GX and10GX; respectively.

For the purpose of characterizing intercellularcoupling and feedback mechanisms, charge movementinto a myocyte via sodium channels QINa and gapjunctions QIInter was calculated by integrating thecurrent flow from the time of upstroke to 95% repo-larization of the fibroblast (Fig. 10). Increasing Rmf

reduced the magnitude of QINa. In the extreme case ofmarginal coupled fibroblasts, QINa was independent ofthe number n. Otherwise, increasing n linearly in-creased the magnitude of QINa. The relationship be-tween n and QIInter was linear for each Rmf. An increaseof Rmf reduced QIInter for each n. Marginally coupledfibroblasts provided an electrical source, otherwisefibroblasts acted as a sink for the myocyte.

Conduction Velocity Modulated by Fibroblasts

Electrical conduction was simulated in a strand of32 myocytes each coupled to a number of fibroblasts.

-100-80-60-40-20 0

20 40

0 0.1 0.2 0.3 0.4

Vm

[mV

]

t [s]

100 ms 20 pA/pF

(c)

-10

-5

0

5

10

15

20

25

30

-120 -100 -80 -60 -40 -20 0 20 40 60

I m [p

A/p

F]

Vm [mV]

[K+]o=5[K+]o=10[K+]o=50

[K+]o=100

(a)

(b)

(d)

(e)

FIGURE 5. Reconstruction of voltage–current relationships of fibroblasts. (a) Voltage steps between -110 and 50 mV wereapplied for a duration of 400 ms from a holding potential of -90 mV. (b) Measured transmembrane currents Im exhibit slowactivation and no apparent inactivation. Data for [K+]o = 10 mM are shown. (c) Simulated transmembrane currents Im reconstructthe measured data. Scales for time and current in (b) and (c) are identical. Stationary currents were extracted for the differentvoltage steps and [K+]o giving (d) measured and (e) simulated voltage–current relationships. (Figures with measured data arefrom43).

SACHSE et al.46

Conduction velocity v and maximal upstroke velocitydV/dtmax were determined (Fig. 11). For a myocyte–myocyte resistance Rmm of 6MX; increasing numbersof coupled fibroblasts decreased conduction velocity vand upstroke velocity dV/dtmax of myocytes.

Conduction velocity v in the strand without coupledfibroblasts was 0.211 m/s (Fig. 11a). In case of 10

coupled fibroblasts, conduction velocity v was reducedto 0.150, 0.163, 0.198, and 0.208 m/s for myocyte–fibroblast resistances Rmf of 10MX;100MX;1GX and10GX; respectively.

Upstroke velocity dV/dtmax in a central myocytewithout fibroblasts coupled to the strand was 128.4 V/s(Fig. 11b). Coupling with 10 fibroblasts caused areduction of the upstroke velocity dV/dtmax to 91.8,99.7, 119.2, and 126.0 V/s for resistances Rmf of10MX;100MX;1GX and 10GX; respectively. The n–vand Rmf–v relationships resembled the relationshipsbetween n and dV/dtmax and Rmf and dV/dtmax,respectively.

Electrical Bridging via Fibroblast Inserts

Electrical bridging between two chains of myocytescoupled via a single fibroblast is demonstrated inFig. 12. The resistance between myocytes Rmm was setto 6MX: A myocyte–fibroblast resistance of Rmf of10MX caused a conduction delay Dt of 2.05 ms(Fig. 12a). A resistance Rmf of 17:416545MX led to themaximal delay Dt of 18.2 ms (Fig. 12b). Conductionwas blocked for resistances Rmf 17:416546MX(Fig. 12c).

The effect of the myocyte–myocyte resistance Rmm

and myocyte–fibroblast resistance Rmf on conductiondelays Dt caused by a single fibroblast is further ex-plored in (Fig. 13). The myocyte–myocyte resistorRmm was varied from 1MX to 100MX: For Rmm largerthan 50MX; conduction was blocked independent ofthe choice of Rmf. For each Rmm otherwise, Dt was anexponential-like monotonical increasing function ofRmf (Fig. 14). The maximal Dt amounted to 20.3 mswith a Rmm ¼ 50MX and Rmf ¼ 48:876354MX: Blockof conduction occurred always for resistances Rmf

larger than the one causing maximal Dt:In order to explore the role of fibroblast’s trans-

membrane currents in the development of conductiondelays, a similar simulation as presented above wasperformed with an ohmic coupling of the myocytechains (Fig. 15). A resistance between these chains Rcc

of 20MX led to a delay Dt between the two centralmyocytes of 2 ms (Fig. 15a). A resistance Rcc of34:908765MX caused the maximal conduction delayof 17.3 ms (Fig. 15b). Conduction was blocked forresistances Rcc 34:908766MX (Fig. 15c).

The effect of the myocyte–myocyte resistance Rmm

and chain–chain resistance Rcc on conduction delays isdemonstrated in Fig. 16. Similar as in simulations withthe arrangement M21–F1–M21, Dt was an exponential-like monotonical increasing function of Rcc for eachRmm and conduction block occurred for resistancesRmf larger than the one causing maximal Dt:

-10

-8

-6

-4

-2

0

-120 -100 -80 -60 -40 -20 0 20 40 60

I Kir

[pA

/pF

]

Vm [mV]

[K+]o=5[K+]o=10[K+]o=50

[K+]o=100

-5

0

5

10

15

20

25

30

-120 -100 -80 -60 -40 -20 0 20 40 60

I Shk

r [pA

/pF

]

Vm [mV]

[K+]o=5[K+]o=10[K+]o=50

[K+]o=100

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

-120 -100 -80 -60 -40 -20 0 20 40 60

I b [p

A/p

F]

Vm [mV]

(a)

(b)

(c)

FIGURE 6. Decomposition of simulated voltage–currentrelationships. The transmembrane current Im is mainly com-posed of two K+ currents: (a) the inwardly rectifying currentIKir and (b) the time- and voltage dependent outward currentIShkr. (c) The contribution of the unspecific background cur-rent Ib is relatively small.


Conduction delays Dt caused by insets of 3 and 5fibroblasts are presented in Fig. 17. Again, the delay Dt

was an exponential-like function of Rmf for each Rmm.Increasing numbers of inserted fibroblasts led todecreasing resistances Rmf leading to maximal Dt:

CONCLUSIONS AND DISCUSSION

Studies of electrical interactions between myocytesand fibroblasts are commonly carried out in cell cul-tures, which provide a controlled extracellular envi-ronment and allow versatile cell arrangement throughappropriate seeding techniques. In this paper, we chosea computational approach to gain insights in theelectrophysiology of coupled myocytes and fibroblastswith particular focus on bridging of conduction be-tween myocytes via fibroblasts.

A mathematical fibroblast model was developedbased on recent voltage clamp, pharmacological and

PCR studies. This model is capable of reconstructingexperimental ion channel activation kinetics and stea-dy-state transmembrane currents for different voltages(Figs. 2 and 5). We used a Nernst and Goldman–Hodgkin–Katz equation based description for themajor currents IKir and IShkr, respectively, to allowreconstruction of measurement data for a wide rangeof [K+]o (5–100 mM). The resting transmembranevoltage was set to -58 mV for [K+]o = 5 mMaccording to measurement data.43 Other studies indi-cated a wide range of resting voltages.23,24 The de-scribed fibroblast model might provide a frameworkfor the description of myofibroblasts, which have beenconsidered to have the same major ionic currents.8

In addition to the fibroblast model, we used thePandit et al. rat ventricular myocyte model34 and anelectrical conduction model to characterize myocyte–fibroblasts interactions. Results from our simulations,in which a single myocyte was coupled to a number of

-80

-60

-40

-20

0

20

40

0 0.02 0.04 0.06 0.08 0.1 0.12V

m [m

V]

t [s]

n=015

10

-80

-60

-40

-20

0

20

40

0 0.02 0.04 0.06 0.08 0.1 0.12

Vm

[mV

]

t [s]

n=15

10

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.02 0.04 0.06 0.08 0.1 0.12

I inte

r [nA

]

t [s]

myocytefibroblast

-6

-5

-4

-3

-2

-1

0

1

0 0.02 0.04 0.06 0.08 0.1 0.12

I inte

r [nA

]

t [s]

myocytefibroblast

0

10

20

30

40

0 0.1 0.2 0.3 0.4 0.5

Na

Cha

nnel

Ava

ilabi

lity

[%]

t [s]

n=015

10 10

20

30

40

50

60

0 2 4 6 8 10

AP

D [m

s]

Number of fibroblasts n

APD50APD90

(a) (b)

(c) (d)

(e) (f)

FIGURE 7. Myocyte–fibroblast interactions simulated with arrangement M–Fn for an intercellular resistance Rmf of 100 MX: Thetransmembrane voltage Vm of (a) a myocyte and (b) fibroblasts were calculated for a number of fibroblasts n varying between 0 and10. Intercellular currents Iinter in case of (c) a single and (d) 10 coupled fibroblasts flows during stimulus and upstroke from themyocyte to the fibroblast(s) followed by reflow at time of the action potential notch and repolarization. (e) Sodium channelavailability was determined as product of the fast and slow inactivation gates. (f) The n-APD50 and n-APD90 relationships arepositive.

SACHSE et al.48

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PD

[ms]


APD50APD90

(b) (c)

(d) (e) (f)

(g) (h) (i)

FIGURE 8. Myocyte–fibroblast interactions simulated with arrangement M–Fn. The number of fibroblasts n was varied between 0and 10. A resistance Rmf of (a–c) 10 MX; (d–f) 1 GX; and (g–h) 10 GX was chosen. The transmembrane voltage Vm of (a, d, g) amyocyte and (b, e, h) fibroblasts as well as (c, f, i) the APDs of the myocyte are shown for various resistances and numbers ofcoupled fibroblasts.

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dV/d

t max

[V/s

]


10 MΩ100 MΩ

1 GΩ10 GΩ

(a) (b)

FIGURE 9. Electrophysiological quantification in arrangement M–Fn. (a) Resting transmembrane voltage and (b) maximal up-stroke velocity were determined for the myocyte. Fibroblasts increase marginally the resting voltage, but modulate significantlyupstroke velocity.


fibroblasts (M–Fn), indicate that coupled fibroblastscan have a significant effect on the dynamics of myo-cyte depolarization (Figs. 7f and 8c, f). However, in allsimulations myocytes developed an action potentialwith peak voltage above 30 mV. In ventricular myo-cytes the depolarization is commonly initiated by in-flow of intercellular or stimulus current causing largesodium currents. We found that the charge associatedwith sodium inflow QINa was an approximately linearfunction of the number of coupled fibroblasts(Fig. 10a). This additional inflow is mostly used tocharge the fibroblast membrane and raise their restingvoltages (Figs. 7b and 8b, e, h). Thus, myocytes wereable to compensate for the additional electrical loadof coupled fibroblasts by increasing sodium channel

current INa. Simulations with two models of guinea pigventricular myocytes30,38 showed similar relationships(data not shown).

A re-flow of this additional charge from fibroblaststo the myocyte occurred during repolarization foraverage and high coupling (Fig. 7c, d). This re-flowreduced repolarization and caused increased APDs(Figs. 7f and 8c, f). In case of marginal coupling,charging of fibroblast membranes was decelerated andstill ongoing during early repolarization. The re-flowstarted in the terminal phase of repolarization.

The charge transported through gap junctionsQIInter from the myocyte to high and average coupledfibroblasts was positive during an action potential(Fig. 10b). Thus, these fibroblasts act as an electrical

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C]


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QIIn

ter [

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1 GΩ10 GΩ

(a) (b)

FIGURE 10. Charge movement in arrangement M–Fn. (a) Charge passing through sodium channels QINa and (b) charge associ-ated with intercellular current QIInter were determined for the myocyte. Negative values indicate summary transport of positivecharge into the myocyte. High and average coupled fibroblasts increase the magnitude of QINa and act as sink for intercellularcurrent.

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/s]


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t max

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FIGURE 11. (a) Conduction velocity in myocyte chain and (b) upstroke velocity simulated with arrangement M32–Fnx32. Theresistance between myocytes Rmm was set to 6 MX: The resistance between myocytes and fibroblasts was varied between 10 MXand 10 GX: Coupling to fibroblasts reduced conduction velocity and upstroke velocity in a strand of myocytes.

SACHSE et al.50

sink. However, the magnitude of QIInter was less than25% of the increase of the magnitude of QINa causedby coupled fibroblasts (Fig. 10a vs. 10b). Thus, morethan 75% of the additional QIInter was associated tothe myocyte and applied at least partly for deceleratedrepolarization. A similarly decelerated repolarization

was not found in our simulations with models of gui-nea pig ventricular myocytes (data not shown).

Our simulations of a strand of myocytes indicatethat increased coupling of fibroblasts can significantlyreduce conduction velocity in cardiac tissue (Fig. 11a).We suggest that this effect is caused by reduction ofmyocyte upstroke velocity (Fig. 11b), which was sim-ilarly visible in our studies of a single myocyte coupledto a number of fibroblasts (Fig. 9b). Decreased up-stroke velocity in the strand studies compared to thosein the single myocyte studies can be explained byintercellular electrical coupling. Repolarized neigh-boring myocytes act as electrical sinks for an excitedcell. Intercellular current flow from a cell undergoingupstroke into depolarized myocytes reduces upstrokevelocity.

Conduction velocity determined in our myocytestrand simulations is small in comparison to velocitiesfrom experimental studies with cardiac tissue of rat44

and other mammals.13,35 Those studies were carriedout at physiological temperature. Experimental studiesof cardiac tissue reported a reduced conductionvelocity at room temperature quantitatively similar asin our simulations.41

Conduction block did not occur in the myocytestrand simulations with 0–10 coupled fibroblasts.However, we expect that for a larger number of cou-pled fibroblasts the conduction velocity would furtherdecrease and block might occur. We suggest that ananalysis applying the safety factor concept42 wouldgive detailed insights in block conditions.

Evidence for bridging of electrical conduction incardiac tissue via fibroblast insets was provided byexperimental studies of cell culture14 and computa-tional studies with a one-dimensional cardiac fibermodel.46 These studies showed reduced conductionvelocity and upstroke velocity in fibroblast insets.Length of the insets was positively correlated with theconduction delay Dt: Large variances of Dt were pre-sented in the experimental study with insets consistingof fibroblasts and also HeLa cells expressing Cx43 gapjunctions.

Our studies of electrical bridging reconstructed theexperimentally found increase of Dt with inset length.The studies were carried out in a one-dimensionalmodel consisting of two chains of myocytes connectedby a chain of fibroblasts. Increased Dt for increasednumber of inserted fibroblasts are demonstrated inFigs. 17b, d and 13b for a wide range of Rmm and Rmf.Maximal conduction delays in our studies were smallerthan in the experimental study, which might beattributed to electrophysiological differences ofneonatal cells used in the cell culture studies and ourmodels describing adult cells. In our studies, conduc-tion block occurred at significantly smaller than

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FIGURE 12. Myocyte–fibroblast interactions simulated witharrangement M21–F1–M21. The transmembrane voltage Vm ofthe central fibroblast as well as its proximal and distal myo-cyte is shown. A myocyte–myocyte resistor Rmm ¼ 6 MX andmyocyte–fibroblast resistor Rmf of (a) 10 MX; (b) 17:416545 MXand (c) 17:416546 MX was chosen, which led to conduction,delayed conduction and block, respectively.


reported values of resistors of myocyte–fibroblast andfibroblast–fibroblast pairs. A possible explanationfor this underestimation is parallel connection offibroblasts in the inset. We expect that using 2- and3-dimensional continuum modeling approacheswill improve the reconstruction of the cell culturestudies. Furthermore, these modeling approaches

would simplify the reconstruction of experiments withmyocyte–fibroblast mixtures in insets and anisotropyof intercellular coupling.

A major goal of our bridging studies was the char-acterization of Dt in relationship to coupling resis-tances and number of fibroblasts. We found anexponential relationship between the myocyte–fibro-blast resistance Rmf and Dt: The value of Rmf formaximal Dt and for conduction block differed by only1X; which presented the resolution limit for the max-ima detection.

An exponential relationship was also found in ourstudies of ohmic coupled two myocyte chains withrespect to the chain–chain resistor Rcc and Dt (Fig. 16).Results of these simulations were very close to theresults with a single fibroblast inset when Rmf was twiceof Rss. These studies revealed that ion channel andcapacitive currents of fibroblasts play a minor role forthe spread of the excitation at least with small numbersof fibroblasts in the inset.

Limitations of the Presented Studies

We did not adopt, develop, or apply descriptions ofmechano-electrical feedback in fibroblasts and myo-cytes. Mechanosensitive currents of fibroblasts have

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FIGURE 13. Conduction delays Dt simulated with arrangement M21–F1–M21. Myocyte–myocyte resistances Rmm and myocyte–fibroblast resistances Rmf were varied. The resulting conduction delays were visualized (a) as three-dimensional plot and (b) withisolines. Each simulated conduction delays is marked with a cross (+).

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FIGURE 14. Exemplary conduction delays Dt simulated witharrangement M21–F1–M21. Delays for myocyte–myocyte resis-tances Rmm of 1, 5, 10, and 50 MX are shown.

SACHSE et al.52

been described in the sino-atrial node andatrium.22,23,27 Mechano-electrical feedback is affectingthe transmembrane voltage and ion concentrations ofcardiac myocytes particularly for high strain and cellswelling.32,39,40 Most of these feedback mechanisms arestill subject of controversies with respect to their

modulation and effects on cardiac tissue in vivo.Commonly, these effects are ignored in computationalstudies of cardiac electrophysiology.

Another limitation of this paper results from theweak experimental characterization of currents otherthan IKir and IShkr in myocardial fibroblasts. Our def-inition of Ib was adapted from models of kidney andcardiac fibroblasts.28,45 The choice of its reversal po-tential Eb is based on experimental studies of rat kid-ney fibroblasts18 and reflects our assumption that thecurrent has several contributors. We expect that futuremodels will decompose Ib in several currents andcomprise intracellular ion handing in more detail asexperimental evidence and quantitative data becomesavailable. Particular candidates are the ubiquitous so-dium–potassium pump, L-type calcium and calcium-activated chloride currents. The latter two candidatesare prominent in rat kidney fibroblasts.18

Our description of fibroblasts neglects steady-stateinactivation of K+ currents, which was reported byShibukawa and colleagues.43 The inactivation wascharacterized by stepping to a membrane voltage of50 mV from holding voltages Vh in the range of -90 to+10 mV applied for 30 s. This steady-state inactiva-tion was negligible for voltages Vh £ -70 mV and notvisible in the clamping protocol with 400 ms voltagesteps from a holding potential of -90 mV (Fig. 5a, b).Thus, only marginal effects of steady-state inactivationcan be expected with the protocols described in ourwork.

Our simulations did not produce evidence for re-flected reentry via fibroblast insets. However, only asmall subset of parameters was varied in these studies.Thus, in general we cannot reject reflected reentry asan electrophysiological phenomenon related to myo-cyte–fibroblast interactions. We suggest that in ourstudies the absence can be explained by electrophysi-ological properties of the sodium channels. Sodiumchannel availability is ruled by fast and slow inacti-vation processes (Fig. 7e). In our simulations theavailability of sodium channels in the proximal myo-cyte was marginal during upstroke of the distal myo-cyte. Reactivation of sodium currents in the proximalmyocyte was prohibited despite significant depolar-ization by currents from the distal myocyte via thefibroblast insert (Fig. 12b). Also the computationalstudies of Vasquez and colleagues did not show evi-dence for reflected reentry despite long conductiondelays up to 50.4 ms [1000]. These studies were per-formed with a model of guinea pig ventricular myo-cytes [1000], which have a long action potentialduration relative to rat ventricular myocytes. Also inthese studies, sodium channel inactivation explains theabsence of reentry.

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FIGURE 15. Bridging simulated with arrangement M21–M21.The myocyte–myocyte resistors Rmm inside of each chainwere fixed to 6 MX: The chains were coupled with a resistor ofRcc of (a) 20 MX; (b) 34:908765 MX and (c) 34:908766 MX;which led to conduction, delayed conduction and block,respectively. The transmembrane voltage Vm of two myocytesproximal and distal to the varied resistor Rcc is shown.


In our simulations, we applied models which de-scribe mean electrophysiological properties of cardiaccells according to recently published experimentaldata.43 As in most computational studies, variabilitiesof properties were not taken into account, despite theseare partly very significant and might severely affect

simulation results. Particularly, resting voltage andmembrane capacity show a large variability in cardiacfibroblasts.23,24,37 We expect that novel stochastic ap-proaches will allow to take these variabilities intoconsideration and thus increase the significance of thisand similar computational studies.15

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FIGURE 16. Conduction delays Dt simulated with arrangement M21–M21 visualized (a) as three-dimensional plot and (b) withisolines. Myocyte–myocyte resistances Rmm and chain–chain resistance Rcc were varied.

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FIGURE 17. Conduction delays Dt simulated with arrangement (a, b) M21–F3–M21 and (c, d) M21–F5–M21. Myocyte–myocyteresistances Rmm and myocyte–fibroblast resistances Rmf were varied. The resulting conduction delays were visualized (a, c) asthree-dimensional plot and (b, d) with isolines.

SACHSE et al.54

ACKNOWLEDGMENTS

This work has been supported by the Richard A.and Nora Eccles Fund for Cardiovascular Research,awards from the Nora Eccles Treadwell Foundation(FBS, APM, JAA), and NIH/NHLBI, HL63969(APM). We thank Ms. Catherine Lloyd, University ofAuckland, and Dr. Alan Garny, University of Oxford,for implementing a CellML version of the fibroblastmodel. We acknowledge the support of Prof. James B.Bassingthwaighte and Dr. Brian Carlson, University ofWashington, for implementation of the fibroblast andmyocyte-fibroblast models in JSim and including thesein the model repository of the NSR Physiome Project(http://www.physiome.org/model/doku.php?id=Cell_Physiology:Fibroblast:model_index and http://www.physiome.org/model/doku.php?id=Cell_Physiology:Action_potential:Myocyte_fibroblast_coupling:model_index).

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Electrophysiological Modeling of Fibroblasts and their Interaction with Myocytes