Electrodiffusion with calcium-activated potassium channels in dendritic spine
EELECTRODIFFUSION WITH CALCIUM-ACTIVATED POTASSIUMCHANNELS IN DENDRITIC SPINE
PILHWA LEE † Abstract.
Feedback of calcium signaling through calcium-activated potassium channels of adendritic spine is investigated. To simulate such ion channels and the resulting spatial distributionof concentration, current, and membrane voltage within the dendritic spine, we apply the immersedboundary method with electrodiffusion . In this simulation method, the permeability to ion flow acrossthe membrane is regulated by the amplitude of chemical potential barriers along the membrane. Withspatially localized ion channels, chemical potential barriers are locally and stochastically regulated.This represents the ion channel gating with multiple subunits, the open and closed states of whichare governed by a continuous-time Markov process. The model simulation recapitulates an inhibitoryaction on voltage-sensitive calcium channels by calcium-activated potassium channels in a stochasticmanner with non-local feedback loop. The model also predicts higher calcium influx with moreclosely placed channel complexes, resolving a potential mechanism of differential calcium handlingby locality of channel distribution. This work provides a foundation for future computer simulationstudies of dendritic spine motility and structural plasticity.
Key words. the immersed boundary method, electrodiffusion, dendritic spine, voltage-sensitivecalcium channel, calcium-activated potassium channel, continuous-time Markov process
AMS subject classifications.
1. Introduction.
Dendritic spines are small protrusions in postsynapse and den-dritic tree of neurons [1], and crucial in learning and memory [2, 3, 4, 5, 6]. In devel-opmental stages, they are highly motile in the course of neuronal wiring and pruning[7, 8, 9, 10]. However, dendritic spine motility and potentially involving structuralplasticity are persistently observed even in matured stages [11] and also in psychi-atric disorders, neuronal injury as well as aging [12, 13, 14]. There are several ionchannels in dendritic spine [15, 16], and relevant modeling with electrodiffusion in themicro- and nano-domains is needful for reconstructing physiological ionic transportsand understanding consequential motility and associated neuronal functions [17].Previously we have studied electrodiffusion with stochastic voltage-senstive cal-cium channels in one-dimensional setting using the immersed boundary method [18].The interface conditions for membrane permeability to each species of ion are replacedby chemical potential barriers in a unified Cartesian domain without explicit dissec-tion of computational domain between intracellular and extracellular domains. In thisarticle, we extend this approach to two-dimensional domain considering spatial effectsof stochastic channel gating and channel distribution. We focus on voltage-sensitivecalcium channels and calcium-activated potassium channels. In the configuration ofspatial expression of those two channels, they are mostly coupled together as “com-plex” with about 15 nm distance in nanodomain [19]. There are two types of calcium-activated potassium channels, small-conductance (SK) and large-conductance (BK)[20]. These potassium channels feed back to voltage-sensitive calcium channels withrepolarization/hyperpolarization to inhibit further calcium influx [21], and regulate inthe synaptic strength and short-term plasticity among many other roles [22, 23, 24].The chemical potential barriers are chosen so that the membrane is permeable toCa and K + locally in space when voltage-sensitive calcium channels and calcium-activated potassium channels are open, respectively. To recapitulate back-propagating † Department of Molecular and Integrative Physiology, University of Michigan, 2800 PlymouthRd, Ann Arbor, MI, [email protected] 1 a r X i v : . [ q - b i o . S C ] S e p P. Lee action potential from soma, the membrane is also regulated uniformly in space to besemi-permeable to Na + for membrane depolarization. The membrane is uniformlysemi-permeable to Cl − all the time of simulation.We place 5 voltage-sensitive calcium channels and 5 calcium-activated potassiumchannels (BK) in the dendritic spine head. The opening and closing of the channelsare modeled by lowering and raising the heights of the chemical potential barriers ofcalcium and potassium according to continuous-time Markov processes. The voltage-sensitive calcium channels have four independent subunits with inactivation from theintracellular local calcium concentration [25]. The channels are open only when allfour subunits are in the open state. The transitions between the open and closed statesof the subunits are governed by voltage sensitive rate constants, and the transition tothe inactivated state is governed by calcium concentration [25]. The calcium-activatedpotassium channels also have four independent subunits with closed and open statesdependent on membrane voltage, and each subunit activated by calcium [25].As a whole, we have incorporated the immersed boundary method with electrodif-fusion for ionic transport in dendritic spine of synapse on 2D, coupled to continuous-time Markov processes for ion channel gating of voltage-sensitive calcium channels andcalcium-activated potassium channels possibly non-uniformly distributed in dendriticspine. The model simulation recapitulates an inhibitory action on voltage-sensitivecalcium channels by calcium-activated potassium channels in a stochastic mannerwith non-local feedback loop. The model also predicts higher calcium influx withmore closely placed channel complexes, resolving a potential mechanism of differen-tial calcium handling by locality of channel distribution.The paper is organized in the following way; in Section 2, the mathematicalformulation of electrodiffusion of ion species, ion-channel gating as a continuous-timeMarkov process, and the resulting regulation of the chemical potential barriers thatmodel ion-channel selectivity are described. In Section 3, two-dimensional study withspatially localized channels are carried out, and the results are presented.
2. MATHEMATICAL FORMULATION.
In this section we consider a fixedtwo-dimensional computational domain with dissolved ions. Immersed within the do-main is a closed membrane, which is fixed in place. The membrane may be permeableor impermeable to each ion species, the permeability being controlled in a gradedmanner by its chemical potential barrier. We have the following notations: D i : diffusion coefficient of the i th ion species q : the unit electrical charge (charge on a proton) qz i : charge of the i th species K B : Boltzmann constant T : absolute temperature (degrees in Kelvin)Ω E : Eulerian domainΩ L : Lagrangian domain ε : dielectric constantThe notations for the variables are the following: x = ( x , x ): Eulerian coordinate x = X ( s ): Lagrangian description ψ i ( x , t ): chemical potential of the i th ion speciesΨ( x ): chemical potential kernel A i ( s, t ) ds : contribution of arc ( s, s + ds ) of membrane to chemical potential of i th ion Fig. 2.1 . Chemcal potential distribution in position and in the course of ion channelgating on 2D : it is placed along the boundary. The support of the chemical potential kernel isa square that measures 24 meshwidths ×
24 meshwidths. The chemical potential is locally andstochastically regulated at the domain of ion channels. (a) The chemical potential for calcium ishigh, i.e. 5 ion channels are all closed. (b) and (c) The chemical potentials for calcium are lowaround 1 and 3 ion channels, respectively. species c i ( x , t ): concentration of the i th ion species J i ( x , t ): flux per unit area of the i th ion species ϕ ( x , t ): electrical potential ρ ( x ): background electrical charge density The chemical potential is expressed in the fol-lowing way: ψ i ( x , t ) = (cid:90) Ω L Ψ( x − X ( s ) , t , n ) A i ( s, t ) ds (2.1)Here, X ( s ) is the configuration of the immersed boundary, where s is a Lagrangianparameter. The function A i ( s, t ) describes the contribution of the membrane at X ( s )to the chemical potential barrier for the i th species of ion. The chemical kernel Ψdefines how the contribution A i ( s, t ) ds is to be spread out in space in the neighborhoodof X ( s ). In the regulation of membrane permeabiliy to the i th species of ion, thechemical potential amplitude A i ( s, t ) is modulated locally or globally. When it islocally controlled, the localized area represents the domain of ion channel for the i th species of ion.In general, any bell-shape function with compact support can be available forchemical potential kernel. For the one-dimensional kernel, a smoothed Dirac deltafunction φ of the second order moment with compact support is used following thefunction suggested by Peskin [28]. The chemical potential kernel in two dimensionaldomain is as follows: Ψ w ( x , t , n ) = 1 w φ ( x · t w ) φ ( x · n w )(2.2)where w is a scaling factor such that Ψ w has a support of a square of edge 4 w . Thecoordinates used in the φ are in the local frame from tangential and normal directions t and n with respect to the membrane at X ( s ). The electrical po-tential is a solution of the Poisson equation: −∇ · ( (cid:15) ∇ ϕ ) = (cid:88) i qz i c i + ρ (2.3) P. Lee where ρ represents the background electrical charge density. The two-dimensionaldomain is prescribed to be periodic in each direction, and the necessary condition forthe existence of the solution of the Poisson equation requires the global electroneutral-ity. In the immersed boundary method with electrodiffusion, local electroneutralityis also satisfied except for the space charge layer around the membrane [26, 27]. ThePoisson equation is solved by Fourier transformation with electrical density given bythe computed concentration for each ionic species. The electrodiffusion equations are for-mulated in the following way: ∂c i ∂t + ∇ · J i = 0(2.4) J i = − D i ( ∇ c i + c i ∇ ( ψ i + qz i ϕ ) K B T )(2.5)Eq.(2.4) is the conservation law (continuity equation) for the i th species of ion. In thisequation c i is the concentration and J i is the flux per unit area of this ion species.Eq.(2.5) gives the flux per unit area as a sum of three terms: diffusion, drift causedby chemical potential, and drift caused by the electrical potential.The membrane voltage, V m ( s ), intracellular concentration, c i ( s ), and ionic cur-rent, I i ( s ) for the ion channel with the center at X ( s ) are collected from the following: V m ( s ) = (cid:90) Ω E (Ψ h ( x − X ( s ) + 2 w n ) − Ψ h ( x − X ( s ) − w n )) ϕ ( x ) d x (2.6) c i ( s ) = (cid:90) Ω E Ψ h ( x − X ( s ) + 2 w n ) c i ( x ) d x (2.7) I i ( s ) = qz i (cid:90) Ω E Ψ h ( x − X ( s ) + 2 w n ) J i ( x ) · n d x (2.8)where 4 w is mostly the width of the membrane, and n is the normal unit vectorat X ( s ) outward from the membrane. In interpolating those quantities with Ψ inEulerian domain, the grid size h is used for regularizing on a compact support. Theelectrodiffusion equations are solved by backward Euler method with a second-orderGodunov upwind method, the details of which are described in [27]. First, let us describe the voltage-sensitive calcium channel gating. The tran-sition between closed and open states of each subunit is expressed as follows:C S α ( V m ) (cid:47) (cid:47) O S β ( V m ) (cid:111) (cid:111) (2.9)where C S and O S represent closed and open states of a subunit. The rate constantsof α and β are functions of membrane voltage V m .We describe the states of subunits and ion channel in on/off way: χ i = (cid:26) S = C i − i − Fig. 2.2 . Markov chain of voltage-sensitive calcium channel : The calcium channel has 4subunits. In the state of C i , i − subunits are open. In the state of I , the channel is in inactivationand closed. The ion channel is open only when it stays on the state of O . The rate constants forthe opening and closing of each subunit are denoted by α and β . Courtesy of Cox [25] .where χ i indicates the open/closed state of i th subunit, and S the state of ion channel.When we express the transition probability between open and closed state of eachsubunit based on Eq.(2.9), P ( χ i ( t + dt ) = 1 | χ i ( t ) = 0) = α ( V m ) dt (2.12) P ( χ i ( t + dt ) = 0 | χ i ( t ) = 1) = β ( V m ) dt (2.13) α ( V m ) = α e q forward V m / . (2.14) β ( V m ) = β e − q backward V m / . (2.15)where α = 3.0 ms − , β = 0.241 ms − , q forward = 1 .
16, and q backward = 1 .
94, where V m is in mV. Eq.(2.12) represents the probability of i th subunit to take transitionfrom closed state at t to open state at t + dt in the infinitesimal time interval dt . Forthe individual ion channel gating, a continuous-time Markov process is applied [29].The ion channel is assumed to have 4 independent subunits; each of them has openand closed states.The channel as a whole is open only when all 4 subunits are in the open state, andwhen the channel is not inactivated. The diagram for the Markov process with discretestates is presented in Fig.(2.2). In the discrete states labeled C i , i − I , the channel is in inactivationand closed. Fig. 2.3 . Markov chain of calcium-activated potassium channel : The channel has 4 sub-units, and calcium binding rate to each subunit is calcium dependent. The rate constants for theopening and closing in 5 different calcium bound states are denoted by k i and k − i , each of whichdependent on membrane voltage. Courtesy of Cox [25] .Next, for the calcium-activated potassium channel gating, the channel is alsoconstituted with 4 independent subunits. Each subunit is activated by calcium, with P. Lee binding and unbinding rate constants K o [Ca] and K − o in the open state, and K c [Ca]and K − c in the closed state. The constants K o , K − o , K c , and K − c are 1.0 nM − s − ,1.065 ms − , 1.0 nM − s − , 11.917 ms − , respectively. With i subunits activated bycalcium, the open and closed states are determined by forward and backward rateconstants, K i and K − i : K i ( V m ) = α ,i e q forward V m / . (2.16) K − i ( V m ) = β ,i e − q backward V m / . (2.17)where α , through α , are 5.5, 8,0, 2.0, 884, 900 s − , and β , through β , are8.669, 1.127, 0.0252, 1.013, 0.1257 ms − , respectively. The rate constants q forward and q backward are 1.16 and 1.94. The numerical algorithm for the ion channel gating isbasically based on the Monte Carlo method, determining first whether to transit, andsecondly the state to transit if needed. The details are described in [18]. As described in the previous section,the continuous-time Markov process for voltage-sensitive calcium channel is appliedwith the membrane voltage and intracellular calcium concentration adjacent to eachion channel, and the chemical potential for Ca is modulated in the on/off way withthe channel state variable S from the Markov process. In the two-dimensional scheme,the chemical potential of calcium is locally regulated. Let the width of the ion channelbe w ch . For the j th ion channel in the state S j , with the Lagrangian parameter s j forthe location of the center of the channel, we do spatial regulation of chemical potentialin the following way: A Ca ( s, t ) = (cid:26) A Ca , open S j = O , s ∈ [ s j − w ch / , s j + w ch / A Ca , closed otherwise(2.18)where A Ca , closed and A Ca , open are specified chemical potential amplitudes to makethe membrane mostly impermeable or semi-permeable to calcium ions (Table 3.2). Forthe domain of membrane without ion channels, the chemical potential for calcium isfixed. The regulation for calcium-activated potassium channels are mostly the same,but the regulated chemical potential is of potassium, and not calcium ionic species.
3. RESULTS AND DISCUSSION.
The membrane changes its permeabil-ity to Ca and K + in a voltage- and calcium-dependent manner according to thecontinuous-time Markov process described above with spatially localized ion chan-nels. For 2D simulation, a periodic square domain with dimensions 2 µ m × µ m iscovered by a Cartesian grid containing 256 ×
256 points with the uniform grid size h in each direction. The model dendritic spine has a diameter 1 µ m and is centeredwithin the computational domain. A timestep ∆ t = 30 µ s is used for all computations,including both the electrodiffusion and the Markov process that controls the openingand closing of membrane channels. For the simulation, 5 voltage-sensitive calciumchannels and 5 calcium-activated potassium channels are uniformly distributed on theupper half circle of the head representing a postsynaptic density (PSD) with 15 nmdistance between two types of channels as a complex. These complexes are labeledindices from 1 to 5 counter-clockwise. Four ion species and background charge areconsidered with initial concentrations different between extracellular and intracelluardomains as shown in Table (3.1).The membrane is stochastically permeable to calcium by the voltage-sensitivecalcium channel. The amount of influx through the channel is regulated by controlling Table 3.1
Initial concentrations in all simulations (mM) . X − denotes fixed background charge and theconcentrations stated for X − refer to the concentration of background charges, not the concentrationof the molecules that carry the background charges. Ion species extracellular concentration intracellular concentrationCa −
150 13Na +
150 15K + − A Ca , open and A Ca , closed as prescribedin Table (3.2). Similarly, the membrane is stochastically permeable to potassium bythe calcium-activated potassium channel. The amount of influx through the channelis regulated by controlling the chemical potential with two parameter of A K + , open and A K + , closed .With the membrane voltage in the range of depolarization, there comes calciumionic influx from the extracellular domain. The depolarization of the membrane isimplemented by lowering the chemical potential of sodium to 42.7422 K B T at t=3ms. The membrane is repolarized by raising the sodium chemical potential to theinitial level, 53.4278 K B T at t=21 ms. The locally regulated chemical potential ofCa around the area of ion channels are shown in Fig.(2.1) with its dynamic changesin amplitude.The calcium concentration distributions in y-section are shown when calciumions are flowed in, and does electrodiffusion at t=4.8 ms and at t=5.4 ms (Fig.3.1(a)). The calcium concentration distribution on 2D are also shown zoomed in att=4.8 ms in Fig.(3.1) (b). The electrical potential distributions in y-section and 2Dare shown at t=7.8 ms and t=8.4 ms (Fig.3.2). At t=7.8 ms, two voltage-senstivecalcium channels are open, and the associated chemical potential is locally lowered.Accordingly the membrane is locally more depolarized showing two spikes. At t=8.4ms, a calcium-activated potassium channel is open, and the corresponding chemicalpotential is locally decreased. There follows a huge upstroke of electrical potentialadjacent to the open BK channel, and the expected membrane repolarization doesappear far away non-locally unexpectedly.The ionic current and membrane voltage in the time course of channel gatingare presented in Fig.(3.3). When the membrane is depolarized and the channels areopen, calcium currents are evidently observed. Interestingly even when the membraneis repolarized, it takes a while for the calcium channels are all closed. The stochasticcalcium-activated potassium channel 1 is shortly open around t=4 ms before calciumconcentration are elevated enough. The evident response of calcium-activated potas-sium channel after calcium inflow by voltage-sensitive calcium channels is shown forcomplexes 3 and 5 with some persistent outflow of potassium ionic current. Here oneinteresting aspect of membrane voltage change is that local membrane voltages are increased around complexes 3 and 5 (red circled), and decreased around complexes 1and 2 (blue circled). This shows that the ionic current from BK ion channel gating isinfluential non-locally with fast redistribution of electrical potential by electrodiffusionas shown in Fig.3.2 (c) and (d).Finally we have tested whether different distribution of complex channels generatedifferent calcium influx, and whether BK channel inhibition drives more calcium in- P. Lee flux. We compare the averaged intracellular calcium concentration at t=30 ms for thenon-uniformly distributed case with that of channel complexes uniformly distributedin the whole dendritic spine head (Fig. 3.4, first two datasets). The two-sample t-test shows a significantly higher calcium level for the non-uniformly distributed case.It is supposed that calcium influx induces membrane depolarization and this pro-vides a positive feedback to adjacent voltage-sensitive calcium channels for them tohave higher probability to open channels. This interaction is thought to be strongerthan repolarizing effects from adjacent calcium-activated potassium channels. Basedon the model simulation result of Fig. 3.2 (c) and (d), once intracellular calciumconcentration around a voltage-sensitive calcium channel is elevated when the chan-nel is open, the adjacent calcium-activated potassium channel gradually responds byopening the channel. What happens here is local depolarization and non-local repolar-ization. Thereafter, this might enhance the coupled voltage-sensitive calcium channelto stay in open state, and the only action of channel inhibition is by calcium-sensitiveinactivation in the channel itself.We also compare the intracellular calcium concentrations with two cases of BKchannels expressed with their non-uniform distribution or knocked out (Fig. 3.4,second and third datasets). The paired-sample t-test shows a significant elevationof calcium influx by BK channel inhibition or knock-out. The significant differencein calcium concentration shows the inhibitory mechanism on calcium influx by theinteraction purely between voltage-sensitive calcium channels and calcium-activatedpotassium channels.
4. Conclusions.
By proposing the immersed chemical potentials , we realize theregulation of membrane permeability to each species of ion, and consequently ion se-lectivity of membrane. With the continuous-time Markov process in the stochasticfeature of the voltage-sensitive ion channel, the ion channel gating and the correspond-ing ionic current are observed dependent on the membrane voltage. In the chemicalpotential barrier landscape for the ion channel, there are several local minima by anumber of subunits of the ion channel. In our approach, that kind of feature is em-bedded in the finite discrete states of the Markov chain, and bell-type shape withoutlocal minima is applied for the chemical potential. The shape of these chemical po-tential barriers also influences the current-voltage relationship of the membrane, theinvestigation of which remains for the future.In two dimensional study, the stochastic ion channel gating with discretely placedion channels were considered. Individual ion channel gating is observed with spatio-temporally changing chemical potential distributions. In comparison to the pointcell modeling, this kind of approach makes it available for us to consider spatially nonuniform distribution of ion channels and their spatial effects on the physiology.Accordingly, in the sense of synaptic plasticity, we can implement the change in theion channel density.The non-local interaction between voltage-sensitve calcium channels and calcium-activated potassium channels is thought to be captured by treating the electrophysiol-ogy in nano- to micro-scale electrodiffusion, and surely not feasible by drift-diffusion.At the same time, positive feedback among local channel complexes for calcium in-flux is also predicted for further validation by experiments. Currently the back-propagating action potential was treated by one-time event of phasic train. In realneuronal activities, the frequency of this back-propagating action potential spikesmay induce differential response from calcium-activate potassium channels [30, 31].Of course, we expect to reconstruct 3D electrodiffusion [32, 33] with the formalismwe exposed here.
Acknowledgements.
The author appreciates careful comments from CharlesPeskin.
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Chemical potential (K B T) prescribed for each ionic species.
Ion species chemical potential high chemical potential lowCa − + + − Fig. 3.1 . Concentration distribution with ion channels open on 2D ; (a) The calciumconcentration distribution in y-section at t=0 ms (red), t=4.8 ms (black-dotted), t=5.4 ms (black-solid). The chemical potential distribution is also shown (blue) at t=4.8 ms. (b) The calciumconcentration is zoomed in to resolve the low level concentration in µM in intracellular domain att=4.8 ms. P. Lee
Fig. 3.2 . Electrical potential distribution on 2D ; (a) The electrical potential in y-section att=0.2 ms (red) and t=7.8 ms (black). The chemical potential for calcium is also shown in y-sectionat t=7.8 ms (blue). (b) The electrical potential on 2D at t=7.8 ms. (c) The electrical potentialin y-section at t=0.2 ms (red) and t=8.4 ms (black). The chemical potential for potassium is alsoshown in y-section at t=8.4 ms (blue). (d) The electrical potential on 2D at t=8.4 ms. Fig. 3.3 . Voltage-sensitive calcium channel and calcium-activated potassium channelactivities ; The columns represent (a) calcium ionic currents by voltage-sensitive calcium channels,(b) potassium ionic currents by calcium-activated potassium channels, (c) membrane voltage, and(d) intracellular calcium concentration adjacent to each voltage-sensitive calcium channel. Therows represent 5 individual complexes of voltage-sensitive calcium channel and calcium-activatedpotassium channel. These electrophysiological activities are from non-uniform distribution channelcomplexes. P. Lee
Fig. 3.4 . Calcium ion concentration comparison ; 10 samples are collected for compari-son of averaged intracellular calcium ion concentrations for three cases, 1) ion channel complexesare uniformly distributed, 2) ion channel complexes are non-unformed distributed, 3) ion channelscomplexes are non-uniformly distributed and BK channels are knocked out. The activity of BKchannel shows an inhibiting function in calcium inflow. Non-uniform distribution of ion channelcomplexes induce higher intracellular calcium concentration. * p < <<