MODELING CALCIUM DYNAMICS IN DENDRITIC ... - CiteSeerX

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SIAM J. APPL. MATH. Vol. 0, No. 0, pp. 000–000

MODELING CALCIUM DYNAMICS IN DENDRITIC SPINES∗ D. HOLCMAN† ‡ AND Z. SCHUSS§ Abstract. Dendritic spines are microstructures located on dendrites of neurons, where calcium can be compartmentalized. They are usually the postsynaptic parts of synapses and may contain anywhere from a few up to thousands of calcium ions at a time. Initiated by an action potential, a back-propagating action potential, or a synaptic stimulation, calcium ions enter spines and are known to bring about their fast contractions (twitching), which in turn affect calcium dynamics. In this paper, we propose a coarse-grained reaction-diffusion (RD) model of a Langevin simulation of calcium dynamics with twitching and relate the biochemical changes induced by calcium to structural changes occurring at the spine level. The RD equations model the contraction of proteins as chemical events and serve to describe how changes in spine structure affect calcium signaling. Calcium ions induce contraction of actin-myosin-type proteins and produce a flow of the cytoplasmic fluid in the direction of the dendritic shaft, thus speeding up the time course of calcium dynamics in the spine, relative to pure diffusion. Experimental and simulation results reveal two time periods in spine calcium dynamics. Simulations [D. Holcman, Z. Schuss, and E. Korkotian, Biophysical Journal, to appear] show that in the first period, calcium motion is mainly driven by the hydrodynamics, while in the second period it is diffusion. The coarse-grained RD model also gives this result, and the analysis reveals how the two time constants depend on spine geometry. The model’s prediction, that there are not two time periods in the diffusion of inert molecules in the spine, has been verified experimentally. Key words. modedling microstructures, reaction-diffusion equations, dendritic spines, calcium, stochastic dynamics AMS subject classifications. 92C05, 92C17, 35K57 DOI. 10.1137/S003613990342894X

1. Dendritic spines and their function. Dendritic spines are microstructures, about 1 µm across, made of a head and connected by a cylindrical neck to the dendrite. Although discovered more than 100 years ago by Ram´ on y Cajal [1] on dendrites of most neurons, including cortical pyramidal neurons and cerebellar Purkinje cells [2], their function is still unclear. The current consensus is that the main function of dendritic spines is to compartmentalize calcium [3]. Regulated by synaptic activity, spines are constantly moving and changing shape [4]. The 100,000 to 300,000 spines on a single spiny neuron drastically increase the active surface of a dendrite [5], [6], and more than 90% of excitatory synapses terminate on dendritic spines. Spines are considered to be basic units of dendritic integration [7], [8], though their role and function are still unclear. There is evidence that morphological changes in spines are associated with synaptic plasticity [4], that is, with the structural and biochemical changes in spines, dendrites, and neuronal synapses. ∗ Received

by the editors June 4, 2003; accepted for publication (in revised form) June 15, 2004; published electronically DATE. http://www.siam.org/journals/siap/x-x/42894.html † Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 Israel (holcman@ wisdom.weizmann.ac.il). The research of this author was supported by the Sloan Foundation and the Schwartz Foundation. ‡ Keck Center, Department of Physiology, UCSF, 513 Parnassus Ave, San Francisco, CA 94143 ([email protected]). § Department of Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel ([email protected]). The research of this author was partially supported by grants from the US-Israel Bi-National Science Foundations, the Israel Science Foundations, and DARPA. 1

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D. HOLCMAN AND Z. SCHUSS

A debate is still raging about the specific function of dendritic spines. In particular, two main views prevail [9], [10]. The first maintains that a dendritic spine constitutes a privileged location for calcium restriction, and consequently, it is a place where synaptic plasticity can be induced. Calcium in dendritic spines triggers changes, such as long-term potentiation (LTP) and long-term depression (LTD) [11], which result in a permanent modification of the synaptic weight. Indeed, calcium dynamics, defined as the rise and duration of concentration inside a dendritic spine, is believed to be determinant for the nature of spine synaptic plasticity. These processes constitute the implementation of some of the memory in the brain at the cellular and subcellular levels. The second view maintains that by changing the shape of the spine, the electrical characteristics of the spine change, thereby modulating the voltage and the depolarization of the dendrite. This way the spines participate in the dendritic computation process. Recently, it has been observed [12] that after calcium ions flow in, a dendritic spine can change shape in a few hundreds milliseconds. This fast change of shape decreases the volume of the spine head. Spine motility was proposed by Blomberg, Cohen, and Siekevitz [13] and the fast twitching movement of the spine was anticipated by Crick in [14], where questions were asked about the rules “governing the change of shape of the spine and, in particular, the neck of the spine” and also on “how these rules are implemented in molecular terms.” Many models of calcium dynamics in dendritic spines have been proposed in the literature [5], [7], [15], [16], [17], [18], [19]; however, calcium dynamics was not considered in conjunction with Crick’s questions and with the observations of [12]. When the spine shape is described by a spherical head connected to a cylindrical neck, several classes of shapes can be distinguished, according to three independent geometrical parameters (see Figure 1). According to this representation, the three parameters are the radius of the head (R), the length of the neck (l), and its diameter (d). There are at most eight possible classes of spines, according to the relative sizes (large or small) of the three parameters. It is not clear yet what are the rules, if any, of the distribution of the different classes in a given neuron. Spines may appear isolated or in clusters on a dendrite [4]. The number of spines and their distribution are regulated by neuronal activity, because increased activity tends to increase the production of spines, whereas light deprivation tends to reduce the number of spines. However, the details remain unclear. Dendritic spines can change shape on various time scales. On the time scale of minutes, synaptic stimulation can generate new spines. LTP experiments in the dentate gyrus are correlated with a change in the diameter of the spine neck. A single spine can split into two, and transitions between filopodia (spines with no head) and the standard form have been observed experimentally [4]. Modulation of sensory inputs, such as monocular deprivation in specific periods of development, modulates spine motility [20], [21]. Spines are less motile in adult neurons than in neurons of juvenile animals. Changes of shape on the time scale of minutes are due to actin (de-)polymerization and can be induced by a variation in the concentration of a neurotransmitter, such as glutamate [12], [23]. Dendritic spines have been observed to move on a very short time scale. For example, vibrations along the spine axis, which are independent of the calcium concentration, occur on the time scale of tens of milliseconds [12]. Spine movements on the time scale of seconds have been observed directly by recent imaging techniques, such as confocal microscopy or two-photon microscopy. It was reported that spines are constantly changing shape [18], [24]. This motility is also

MODELING CALCIUM DYNAMICS IN DENDRITIC SPINES

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channels part 1 Spine head

Calcium ions ActinMyosin type molecules

part 2 pumps part 3

beginning of the spine neck Spine neck

part 4

Pumps spring model of the actin network

Dendrite shaft

Calcium ions ActinMyosin type molecules Ionic channels

Fig. 1. Schematic description of a dendritic spine. A dendritic spine is modelled as a spherical head connected to the dendrite by a cylindrical neck. The surface of the head contains a postsynaptic density, where various types of protein channels, such as the glutamate receptors NMDA and AMPA, are anchored and conduct ions into the spine when opened. A spine contains signalling molecules, such as calmodulin; cytoskeletal proteins, such as actin-myosin; and organelles, such as smooth endoplasmic reticulum. Pumps are located on the side and channels at the top of the spine head. Actin-myosin sites are represented schematically, attached to the actin network, so that a contraction of a single protein affects the entire spine. (Figure reprinted from [D. Holcman, Z. Schuss, and E. Korkotion, Calcium dynamics in dendritic spines and spine motility, Biophysical Journal, 87 (2004), pp. 81–91.] with permission.

an actin-dependent process. The postsynaptic current can be modified by affecting the spine geometry, thus modulating the synapse. Changes in spine shape can then affect the efficacy of calcium dynamics. Specifically, it was reported in [18] that changing the spine neck affects the time course of calcium dynamics: High calcium concentration is maintained for a shorter period of time when the neck is shorter. Thus dendritic spines with shorter necks are less efficient in compartmentalizing calcium. In summary, spines undergo a constant readjustment, which can be viewed as an intrinsic spine property [20], and motility possibly contributes to synaptic plasticity. It has recently been observed [12] that a dendritic spine can change shape quickly, on the scale of a few hundreds of milliseconds, after calcium ions flow into the spine. Transient calcium causes the spine to twitch (see [12]). This quick change of shape consists in head contractions oriented on the average in the direction of the dendritic shaft. This contraction can be induced by agonists, such as a neurotransmitter, or by a back-propagating action potential. Evidence of high concentration of actin and myosin in spines was reported in [25], where proteins were observed to form clusters inside the spine head, near the channels. These clusters are called the postsynaptic density (PSD). More uniform distributions of clusters of myosin molecules were also observed. As in muscle cells, high concentrations of actin molecules indicate that rapid movement can be ascribed to the contraction of these molecules, because blocking them prevents all shape fluctuations [12].

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It is remarkable that the description of the diffusive motion of ions in spines can be considered in the intermediate regime between continuum and discrete. Due to its specific geometry, the dendritic spine can be studied as a separate unit from the remaining part of the dendrite. Chemical reactions in the spine involve only a small number of molecules (hundreds), which explains the relatively large fluctuations in the reactions. These may lead to synaptic plasticity. This fact also reinforces the idea that the spine has a major role in converting a random signal, carried by the motion of ions or secondary messengers, into a more deterministic, less fluctuating, and more stable variable, represented by the synaptic weight. 2. Introduction. Despite the rapid development of high-quality technology, today’s biophysical analysis of calcium in dendritic spines is limited by the resolution of the instruments. Thus models become useful tools for the analysis and prediction of spine activity, based on the evidence of molecular chemical reactions. 2.1. Modeling the dendritic spine dynamics. We propose here an answer to Crick’s question about the cause and effect of the twitching and its role in the functioning of the spine as a conductor of calcium. Specifically, we attribute the twitching motion to the contraction of actin-myosin-type proteins, denoted AM, when they bind calcium, and include its effect on the dynamics of the calcium ions in the spine. This is the first quantitative theoretical and mathematical treatment of the twitching and its role in calcium dynamics in the spine. In [26] we constructed a Langevin dynamics simulation of calcium dynamics in the spine, and here we propose a continuum model of the same. The calmodulin proteins (CaM) can bind up to four calcium ions to form the complex CaMCa4 . This complex starts other important chemical reactions, involving, for example, calmodulin-protein kinase-II. This kinase plays a crucial role in LTP induction [16]. When a sufficiently large number of CaMCa4 complexes are formed, it produces LTP changes and/or induces dephosphorylation and (de-)polymerizations. It can also affect certain biophysical properties of certain channels, such as N-methylD-aspartale (NMDA) receptors. More generally, this type of reaction is known to induce modifications in the spine shape and biophysical changes at various levels, such as synaptic modifications, and changes in the number of channels (see [17]): When channel subunits are modified, the selectivity and/or ionic conductivity is changed, affecting the number of ions that enter the spine. When the number of receptors increases, e.g., of AMPA receptors, the spine’s depolarization increases, resulting in a higher probability of opening of NMDA receptors and thus increasing the total number of calcium ions entering the spine. We model the spine as a machine powered by the calcium it conducts, and we describe here the induced movement. Proteins involved in the calcium conduction process are found inside the dendritic spine. Their spatial distribution was reported in [25]. As mentioned above, relevant proteins involved in spine motility include actin, which has been shown in [12], [20], [24] to be directly involved in the biophysical process underlying fast spine motility. We maintain here that AM sites are driving the motility events. It was shown in [25] that dendritic spines contain a network of myosin molecules. The spatial distribution of myosin molecule in the spine has been observed to be uniform and to be sparse inside the PSD. From a biological point of view, it is of primary interest to answer two related questions about calcium dynamics in dendritic spines, after their channels open: (1) How much calcium is there inside the spine? (2) How long does a given quantity of calcium stay inside the spine? Obviously, the answers depend on the geometry of the


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cell. In this context, the aim of our model is to reproduce the time course of events, such as calcium dynamics, which determine the transition between depression and facilitation, or long- and short-term depression (see [28]). We propose that calcium ions set the machine in motion by initiating the contraction of AM as they bind at active sites [26]. We elucidate the cause and effect of twitching in the functioning of the spine by adding up the local contractions of the separate calcium-saturated proteins to achieve a global contraction effect. The contraction of the spine head induces a flow field of the cytoplasmic fluid, which in turn pushes the ions, thus speeding up their movement in the spine. 2.2. Biological consequences. We reported and discussed the biological consequences of a Langevin simulation, designed at a molecular level, in our first paper [26]. The purpose of the simulation was to investigate at a molecular level biochemical events induced by calcium and thus to explain structural changes occurring at the spine level. The main biological conclusion of [26] concerns the quantification of the effect of the hydrodynamical push on calcium dynamics in the spine. In particular, we showed not only that the push effect is created by the calcium ions, but that the push targets the same calcium ions towards the dendrite and in the direction of the center of the spine, where the spine apparatus and other relevant proteins are located. The flow due to the push does not allow the calcium ions to stay inside the spine head and to return to the head once they are inside the neck. The drift increases the efficiency of calcium conduction from the synapse to the dendrite and speeds up the calcium clearance of the spine. The simulations of [26] show that in the absence of the drift effect, the proportion of calcium ions conducted to the dendrite is two to three times smaller than in its presence. This led to the prediction that there are not two time periods in the diffusion of inert dye molecules in the spine, as has been recently verified experimentally [27]. We propose here a coarse-grained description of the coupling between changes in spine structure and calcium dynamics. A set of nonlinear reaction-diffusion equations is derived from the Langevin description. The analysis of the model reveals the time scale of the hydrodynamical effect and leads to the calculation of the time constant of the first concentration decay period. Consequently the push effect offers a possible reinterpretation of the results of [19], about the double exponential decay of the calcium concentration inside the spine. The first decay period was reported in [19] to be the consequence of buffered calcium diffusion from spine to dendrite and the effect of calcium pumps in the spine head. According to [19], the second period starts when near equilibrium is achieved between the spine’s and the dendrite’s calcium concentrations. We show in this paper that the two time periods of calcium concentration decay are recovered, under specific conditions, when the hydrodynamical push effect is included. We observe that the decay, corresponding to a predominantly hydrodynamical effect, starts immediately after the ions enter the spine head. This decay is rapid and its duration is random. It ends when hardly any saturated contractive molecules are left. The ionic motion in the second period is mainly pure diffusion and pump extrusion. An analytic expression for the fast decay rate is derived from the model in terms of the average hydrodynamical flow velocity. The main biological result of our model is that the rapid spine movement produces fast clearance of calcium from the dendritic spine and directs calcium ions to a specific location between the neck and the dendritic shaft, preventing the pumping out of the majority of ions. As mentioned above, the model also predicts that there are not two time periods in the diffusion of inert dye molecules in the spine, which means that diffusion alone cannot

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be responsible for the double exponential decay. 2.3. The need of a molecular approach. The models of calcium diffusion in dendritic spines used, e.g., in [5], [6], [7], [15], [16], [17], [18], [19], [22] are based on a phenomenological approach that uses the coupling between the diffusion equation and the ambient chemical reactions. They are based on compartmentalization of the spine into several subunits, where the calcium diffusion process is discretized, while ordinary differential equations describe the chemical bonding of calcium to buffer protein molecules. In this paper, we present a mathematical model of calcium dynamics in dendritic spines, based on molecular-level considerations. Actually, we propose a unified approach to modeling calcium dynamics inside microstructures, including dendritic spines, that postulates Brownian motion of the calcium ions in the cell. The randomness of the ionic motion becomes significant when the number of ions in the cell is small. At the molecular level, all phenomena, beginning with the motion of a single ion and up to the dynamics of the entire ionic population, are stochastic processes. These include the random walk of an ion, forming or breaking bonds with proteins by an individual ion. On the entire calcium population level, they include the dynamics of the number of bound proteins, which depends on the trajectory of each ion, and the distribution of the protein molecules. Our model begins with the description of the dynamics of individual calcium ions in terms of a system of Langevin equations. The collective effect inside the spine of the entire calcium population, due to the interaction between the calcium and the proteins, is captured in our model by a nonlinear drift term that couples the hydrodynamical flow field to the number of ions bound to certain proteins. This produces a new effect that has to be included in the diffusion equation. The distribution of proteins inside the cell becomes an important part of the model. 2.4. Biological simplifications of the model. We make several simplifications in constructing the model of the spine. Thus, we neglect other types of organelles that are also involved in calcium dynamics: the spine apparatus, mitochondria, and other types of proteins. We have included a low concentration (.5–1 µM) of binding molecules such as calcineurin. However, at this concentration these molecules cannot capture fully the role played by the buffer activity. The present model ignores the effect of a large buffer regulation, but we keep in mind that it can affect the calcium dynamics. Furthermore, it is known that calcium stores in the spine release calcium ions when prompted by external calcium ions, under specific conditions. We neglect this effect here to avoid complicating our model. We also restrict the biochemical structure of the spine by singling out the CaM, AM, calcineurin, and one type of calcium pump. All these proteins constrain calcium flow in the dendritic spine by binding calcium ions for random periods of time. The technical assumption in our model is that the AM proteins contract at a fixed rate as long as they keep four calcium ions bound. Thus contraction begins and ends at random times. Since we are interested in the dynamics of calcium, when the ions are already inside the spine, we avoid the computation of the transient time starting from the action potential and the opening of the voltagesensitive calcium channels. The specific geometry of the spine needs to be considered in order to evaluate the time evolution of calcium concentration in the spine. In the present simplified model, the spine geometry has been described by three parameters: the length and diameter


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of the spine neck and the radius of the spine head (see Figure 1), smoothing out the local irregularities of the boundary. Another geometric feature is the distribution S0 (x) of calcium-dependent molecules that contract when they bind enough calcium. Two extreme possible distributions of proteins have been considered in the simulation, reported in [26]: a uniform distribution inside the head or an accumulation at the PSD area and the simulations show that the calcium dynamics depend on the distributions of the proteins. In reality, a mixture of the two distributions is observed in [26], but we will ignore it in the derivation of the decay rate. 3. A simplified physical model of the spine. The two main components of the dendritic spine in our model are a spherical head and a cylindrical neck, which connects it to the dendritic shaft. On top of the spine head, opposite the neck, there are protein channels that conduct calcium into the spine head. These channels can be of two types: NMDA channels (opened by the glutamate neurotransmitter) and calcium channels, which are voltage sensitive. There are only 2–5 NMDA channels open at a time. For the purpose of this model, we use only the location of these channels as the initial positions for the ions. Our model concerns times after the calcium ions have entered the spine head. A schematic figure of the spine is presented in Figure 1. Active pumps are located on the lower half of the spine head. Their role is to conduct calcium out of the spine head. Pumping is an active process that requires energy, provided by the adenosine tri phosphate (ATP) molecules, whereas when calcium enters through the channels, no extra molecular energy is needed. We assume that there is only one ion at a time inside a pump and, due to the active structure, it requires a certain time to be pumped out. This time can be assumed random or deterministic. The latter case is valid when the exit time distribution is concentrated around the mean value. In a coarse-grained continuum model the pumping time is neglected, so the part of the boundary occupied by pumps becomes an absorbing boundary. The many organelles inside the spine head do not affect the nature of the random motion of ions, mainly due to their large size relative to that of ions. They only restrict the volume available for free diffusion of calcium. Neglecting their presence effectively frees the interior of the spine head from obstacles to ionic movement. This can be compensated for by decreasing the radius of the head. The incompressible cytoplasmic fluid that fills out the spine and its flow are a part of our model. 3.1. A schematic model of spine twitching. Once calcium ions enter the spine they reach AM binding sites by diffusion and can bind there. When four calcium ions bind to a single AM protein, a local contraction of the protein occurs. All the local contractions at a given time produce a global contraction and induce a hydrodynamical movement of the cytoplasmic fluid. Calcium ions can reach the dendritic shaft through the spine neck and be totally absorbed there, or they can be pumped out of the spine by active pumps. Our model allows us to calculate the fraction of ions that are pumped out, relative to those that reach the dendrite. At a molecular level, in a phenomenological approach, a contraction produced by AM occurs with a characteristic time tc = 1 ms, say, and the length fluctuates about lc = 0.02µm [2, p. 681], depending only on the type of protein. In a homogenization approximation of the spine head, the result of this local contraction produces a fixed average velocity of the cytoplasmic fluid of the order vQ = tlcc = 0.02µm/ms. Since it is known that there are only a few AM binding sites (less than a hundred [25]), each binding event can modify the dynamics significantly. It is important

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therefore to keep track of the number of bound ions at any given time. Both the distribution of AM binding sites and the binding times are random. Consequently, the twitching of the spine head is also random. This, in turn, implies that the evolution of calcium concentration inside the spine is random. In a continuum description of this process, only average motion is observed, so the random realizations that can be observed in molecular simulations are smoothed out. 3.2. Final model simplifications. We simplify the model further by neglecting the long range ion-ion electrostatic interactions, as well as the ion-protein interactions. At a molecular level, when 500 calcium ions enter the dendritic spine, they create a difference of potential of about 16 mV (compared to −70 mV of the cell potential), so there are enough negative ions inside the spine to electrostatically neutralize the calcium ions. Specifically, the cooperative effect of the ions creates dipoles that screen the long-range interaction forces r12 to short-range interactions. The shield around each ion is a basis for an approximation that neglects the electrostatic forces in order to study the dynamics of calcium ions inside the spine. In this approximation the trajectories of the calcium ions are independent. The ion-water interaction is simplified into hydrodynamical drag and a zero mean fluctuating force that describes the randomness of the water-ion collisions [29]. The ion-protein interaction near a binding site, where a high electrical field targets the ions toward the active center of a binding site, is represented by a short range parabolic potential well. This allows us to include the backward binding reaction constant in the model. The effect of the forward constant is discussed below. Each time an ion nears an active neighborhood of a protein, we assume that the electrostatic forces direct the ion so that a bond is formed with a given probability, depending on the forward rate constant. The backward reaction rate is the reciprocal of the mean time an ion stays bound. A binding site that holds an ion cannot bind additional ions before the bound ion escapes. We say that a protein is saturated if each of the four binding sites contains a calcium ion at the same time. The chemical kinetics of the binding and unbinding of calcium to and from the substrate proteins (CaM, AM, calcineurin) in the spine cannot be described by the usual Arrhenius kinetics because of the small number of the reactant particles, the large fluctuations in the number of bound ions, and the hydrodynamic effect on the binding and unbinding reactions. We describe the forward and backward reactions on a molecular level in section 5 and then coarse-grain the equations in section 6. We consider in our model only two classes of binding proteins: one that includes CaM and AM (that is, proteins that can bind 4 calcium ions) and a second that includes calcineurin, which can bind only one calcium ion at a time. The simplified model described above was used for a molecular simulation of calcium dynamics in a dendritic spine in [26]. 4. The mathematical model. The mathematical model of the simplified physical model of the dendritic spine has several components. First, the domain Ωt , available for the motion of an ion at time t, has quite a complicated geometry, due to the presence of many obstacles, as mentioned in section 3, and it may change in time. Actually, this change is one of the main phenomena captured by our model. Second, when Ωt changes, a flow of the cytoplasmic fluid in the dendritic spine ensues, which in turn gives rise to an hydrodynamical drag force on the ions inside the dendritic spine. This drag is a frictional force proportional to the relative velocity between the ions and the fluid. This force is not neglected in our simplified model. Third, the mathematical expression of these assumptions is a Langevin model of the ionic


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motion. That is, the motion is described by a system of identical uncoupled Langevin equations driven by independent Brownian motions. 4.1. Mathematical simplifications. To simplify the analysis and simulation of the spine, we make several drastic simplifications. The quality of the simplified model is evaluated by its ability to capture the main phenomenology observed in experiment and by its ability to predict the fluid flow and the time dependence of the measured calcium concentration inside the dendritic spine. The first simplification is that we consider the ions to be point charges, that is, we neglect Lennard–Jones repulsion. The second simplification is that we neglect electrostatic ion-ion interactions. This means that we can neglect all ionic species except the calcium, whose concentration needs to be predicted. We replace electrostatic interactions by interactions with a fixed mean field (that is, with a field not computed from Poisson’s equation). Thus, we assume that the calcium ions move in an effective electrostatic field created by their interactions with each other and with other ions and by the permanent charge distribution on the CaM, calcineurin proteins, and AM complex. The behavior of this potential is assumed, rather than computed. We also neglect the change in the shape of the potential when a calcium ion binds to a protein molecule. The third simplification is that we neglect the impenetrable obstacles to the ionic motion posed by the presence of the proteins. Thus, we assume that the ionic motion inside the dendritic spine is geometrically unrestricted. Therefore, the domain Ωt is the interior of the dendritic spine. 4.2. The Langevin equations. For a dendritic spine containing N ions of different species (e.g., Ca++ , Na+ , Cl− , and so on), xi (t) is the displacement vector ˜ = (x1 , x2 , . . . , xN ) is the of the ith ion, mi is its mass, and zi is its valence. x coordinate of the N ions in configuration space. We assume that a flow field V (x, t) is given (see description below) and that ions interact with a fixed potential of the charges on the proteins, U0 (x), and with the variable potential of all other ions. The variable potential consists of both the electrostatic ion-ion interaction potential, Uii (˜ x), and the potential of Lennard–Jones-type repulsions, ULJ (˜ x) (that represents the finite size of the ions). The force per unit mass on the ith ion is F i (˜ x) = −zi e∇xi U0 (xi ) + Uii (˜ x) − ∇xi ULJ (˜ x). The dynamics of the ith ion is given by the Langevin equation ¨ i + γi [x˙ i − V (xi , t)] + F i (˜ ˙ i, x (4.1) x) = 2i γi w where e is the electronic charge. Here i = kB T /mi , T is the temperature, kB is the Boltzmann constant, γi = 6πai ηi is the dynamical viscosity (where ηi is the viscosity coefficient per unit mass), and ai is the radius of the ion. The frictional drag force, −γ [x˙ i − V (xi , t)], is proportional to the relative velocity of the ion and ˙ i represent the thermal fluctuations of the the cytoplasmic fluid. The accelerations w fluid. The relation between the velocity diffusion constant and the friction coefficient, Di =

kB T , mi γi

is Einstein’s fluctuation-dissipation principle [30]. In the Smoluchowski limit of large damping [30], the Langevin equation (4.1) reduces to ˙ i. γi [x˙ i − V (xi , t)] + F i (˜ (4.2) x) = 2i γi w

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In this paper, we neglect the ion-ion interactions; that is, we set ULJ (˜ x) = Uii (˜ x) = 0 so that (4.2) becomes ˙ i, (4.3) γi [x˙ i − V (xi , t)] + F (xi ) = 2i γi w where F (xi ) = −zi e∇xi U0 (xi ). Since we are interested in tracing only one species in the spine, namely, the concentration of calcium, we assume that γi = γCa++ , mi = mCa++ , zi = z = 2. Under these assumptions, equations (4.3) are independent and identical, so that their transition probability densities are identical. We denote the transition probability density function (PDF) of each ion by p(x, t | x0 , t0 ) so that the calcium concentration is p(x, t | x0 , t0 )c0 (x0 ) dx0 , c(x, t) = Ωt

where c0 (x0 ) is the initial calcium density. 4.3. Reaction-diffusion description of the binding and unbinding reactions. We derive a reaction-diffusion system of equations in a slightly more general setting. We consider a single reactant M (e.g., calcium), whose density, cM (x, t), satisfies the Nernst–Planck (or Smoluchowski) equation corresponding to the Langevin dynamics (4.2) [30], (4.4)

∂cM (x, t) = −∇ · J (x, t), ∂t

where the flux J (x, t) is defined as F (x, t) cM (x, t) − D∇cM (x, t). J (x, t) = V (x, t) − (4.5) γ The immobile substrate protein S is represented in this model by the potential U0 (x, t) of the electrostatic force F (x, t). This force varies in time as reactant ions bind to or unbind from the substrate, thus changing the net electrostatic charge on the substrate. Instead of following the details of the binding and unbinding process and the fluctuations in the force F (x, t), we coarse grain the Nernst–Planck equation (4.4) by replacing it with reaction-diffusion equations. To formulate our problem in terms of reaction-diffusion equations, we partition the boundary of the domain Ω into three parts: the pumps and the bottom of the neck, denoted ∂Ωa (t), which absorb calcium ions; the remaining surface of the head, denoted ΣH (t); and the surface of the neck, denoted ∂ΩN , where the normal flux equals the velocity of the boundary at each point. We introduce the variables S (j) (x, t), (0 ≤ j ≤ 4), that represent the number of proteins in a test volume about x that contains j bound M ions at time t. Then the number of occupied binding sites on these proteins is jS (j) (x, t), and the number of free binding sites on these proteins is (4−j)S (j) (x, t). Obviously, at all times 4 j=0

S (j) (x, t) = S0 (x),


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where S0 (x) is the number of proteins in the volume element. We assume that the forward and backward reaction rates, k1 and k−1 , respectively, are constant and independent of the densities (see discussion in section ?? below). It follows that the reaction-diffusion equations for the number of free calcium ions, M (x, t), and S (j) (x, t) are 4 ∂M (x, t) = −∇ · J M (x, t) − k1 M (x, t) (4 − j)S (j) (x, t) ∂t j=0

(4.6)

+k−1

4

jS (j) (x, t),

j=0

(4.7)

∂S (j) (x, t) = k1 M (x, t) (5 − j)S (j−1) (x, t) − (4 − j)S (j) (x, t) ∂t

−k−1 jS (j) (x, t) − (j + 1)S (j+1) (x, t) ,

where the flux is defined by (4.8)

J M (x, t) = −D∇M (x, t) + V (x, t)M (x, t),

and S (−1) (x, t) = S (5) (x, t) = 0. The initial conditions are (4.9)

S (0) (x, 0) = S0 (x),

S (j) (x, 0) = 0

for 1 ≤ j ≤ 4.

The system (4.6), (4.7) is a coarse-grained reaction-diffusion model of the transient chemical reaction in Ω(t). Renormalizing the numbers of the different species per unit test volume converts them into densities. Obviously, the forward rate constant k1 has to be changed accordingly. The initial and boundary conditions for M (x, t) are the initial reactant density, absorption at the absorbing boundary, and flux given by the motion of the reflective boundary, M (x, 0) = c0 (x) (4.10)

M (x, t) = 0 J M (x, t) · ν(x) = 0

(4.11)

∂M (x, t) =0 ∂n(x)

for x ∈ Ω(t),

for x ∈ ∂Ωa (t), for x ∈ ∂ΩN , for x ∈ ΣH (t).

The boundary condition (4.11) means that the boundary flux J M (x, t) · ν(x) (see (4.8)) is actually the flux of the particles carried by the moving boundary. Note that V (x, t) · ν(x) = 0 on ∂ΩN . The geometrical effect of substrate distribution is expressed in S0 (x). There are no moving internal boundaries, because the support of S (j) (x, t) at all times is that of S0 (x). 4.4. Specification of the hydrodynamical flow. The flow of the incompressible cytoplasmic fluid, as explained above, is due to the local contraction of the saturated AM complexes. We assume that the flow field is derived from a potential φ(x, t)

12


(see, e.g., [31]), V (x, t) = ∇φ(x, t).

(4.12)

The incompressibility condition, ∇·V (x, t) = 0, reduces to the Laplace equation in the head ΩH (t) of the spine at time t. The surface of the head, Σ(t), is partitioned into the surface ΣH (t) of the spine head that does include the surface common with the neck and the cap ΣN (t) of the surface of the head inside the neck, Σ(t) = ΣH (t) ∪ ΣN (t). The Laplace equation in ΩH (t) is (4.13)

for y ∈ ΩH (t), t > 0 ,

∆y φ(y, t) = 0

with the boundary conditions ∂φ(y, t) (4.14) = −V (t), ∂n y∈ΣH (t)

∂φ(y, t) = F (V (t)), ∂n y∈ΣN (t)

where V (t) is the average velocity induced by the deformation of the head (see (4.17) below and the appendix), due to the sum of all the local contractions, and F (V (t)) is the induced field velocity at the top of the neck ΣN (t). The function F (V ) is described in the appendix. The quantities V (t) and F (V (t)) are stochastic processes that are proportional to the number of saturated proteins at any given time t. The flow field can be expressed explicitly in terms of the functions V (t) and F (V (t)) by Green’s function for the Neumann problem for Poisson’s equation in a sphere (or a disk) through Stokes’s formula. Green’s function G(x, y, t) is the solution (defined up to a constant) of the equation −∆y G(x, y, t) = δ(x − y) −

1 |Ωt |

for x, y ∈ ΩH (t),

(4.15) ∂G(x, y, t) =0 ∂ν(y)

for x ∈ ΩH (t), y ∈ Σ(t).

Multiplying (4.13) by G(x, y, t) and (4.15) by φ(y, t) and integrating with respect to y over the domain, using Stokes’s theorem and the boundary condition (4.14), we get ∂φ(y, t) ∂G(x, y, t) φ(x, t) = G(x, y, t) dSy − φ(y, t) dSy ∂n ∂n y∈Σ(t) y∈Σ(t) 1 + φ(y, t)dy VH ΩH (t) ∂φ(y, t) 1 G(x, y, t) dSy + = φ(y, t)dy ∂n VH ΩH (t) y∈Σ(t) =− V (t)G(x, y, t) dSy + F (V (t))G(x y, t) Sdy ΣH (t)

+

1 VH

ΩH (t)

φ(y, t)dy

= −V (t) 1 + VH

ΣN (t)

ΣH (t)

ΩH (t)

G(x, y, t) dSy + F (V (t))

φ(y, t)dy.

ΣN (t)

G(x, y, t) dSy


The flow field is given by ∇φ(x, t) = −V (t)

ΣH

13

∇x G(x, y) dSy + F (V (t))

ΣN

∇x G(x, y) dSy .

In the neck, due to the symmetries and the uniform initial conditions, we simplify the flow field by assuming its velocity is parallel to the axis of the neck. It is given by ∇φ(x, t) = V (x, t) = F (V (t))k, where k is a unit vector along the axis of the neck. In order to close the equations, we recall that the velocity of the boundary V (t) is a function of the number of proteins S (4) (t) = (4.16) S (4) (x, t) dx Ω

that are saturated at time t. Thus V (t) = vQ S (4) (t), (4.17) F (V (t)) = KvQ S (4) (t), where vQ is a constant velocity, depending on the nature of the contractile protein, and K is a constant that depends on the geometry of the spine and the dimension (here 2 or 3). We note that, according to (4.17), as the number of saturated proteins increases the hydrodynamical flow begins to dominate the diffusion. 5. The chemical kinetics of the binding and unbinding reactions. The k forward binding reaction of M to S, M + S 1 M S, is governed by a forward rate constant k1 , because the process of binding consists of an ion falling into a potential well. The survival probability of a single ion inside the spine head, in the presence of potential traps, decays exponentially fast, so that the rate constant for binding is the exponential decay rate. If the binding process involves many ions, the binding rate is the total absorption flux on the boundaries of the potential wells [32]. More precisely, the instantaneous binding rate is

k1 (t) = (5.1) J (x, t) · ν(x) dSx , ∂ΩS(t)

where ∂ΩS(t) is the boundary of the free binding sites on the substrate at time t. An approximation to k1 (t) can be obtained by replacing the flux density J (x, t) · ν(x) with its instantaneous average over the entire boundary ∂ΩS(t) . Then the local instantaneous binding rate of calcium near x is (5.2)

k1 (t) = k1

4

(4 − j)S (j) (x, t),

j=0

where k1 is the forward binding rate per ion per protein and S (j) (x, t) is the number of proteins with j attached calcium ions. When the radius of a potential well with circular cross-section is Rp , the forward binding rate constant k1 is given by Smoluchowski’s formula (5.3)

k1 = 2πRp DM ,

14


where DM is the diffusion constant of M -ions [32], [33]. This determination is done for a separate reaction, not necessarily in the domain Ω. The forward rate constant k1 is an input parameter into the model, e.g., from a molecular dynamics simulation or from direct measurement in a separate chemical reaction [26]. Note that the forward binding rate depends on the radius of the potential well, but not on its depth. The backward binding rate, k−1 , is the rate at which ions escape the potential well. According to Kramers’s theory [30], [34] such a dissociation is due to thermal activation of the ions inside the potential well, and its rate is given by the Arrhenius law with a given activation energy. We recall that in Kramers’s theory of thermal activation over a smooth (parabolic) potential barrier, the dissociation rate is onehalf the reciprocal of the mean first passage (MFPT) of an ion, initially inside the well, to its boundary [30], [34], [35]. This constant is also an input parameter. Given k1 , k−1 , both the depth and the radius of a binding site can be selected by calibration according to (5.3) and Kramers’s formulas. Explicit calculations are given in [36]. 6. Simulation of calcium kinetics in dendritic spines. When channels open, the maximal number of calcium ions that flow into the dendritic spine is of the order of a few hundred [19], which is also the order of magnitude of the number of calmodulin or myosin molecules inside the spine head. A Brownian simulation gives a description of calcium dynamics over a wide range of parameters, starting with only one ion in the spine and up to a number, where a continuum approximation is valid. In such a simulation the number of bonds formed by each calcium ion can be monitored over time. The number of bound proteins at a given time is a random process, because the forward and backward binding processes occur at random times and at random places. Consequently, the twitching movement of the dendritic spine is a random process as well. Simulations of our model give the probability that an ion forms a bond, given the protein distribution. They also demonstrate the role of the drift in modifying the recurrent bindings and unbindings of the Brownian particles to given proteins. 6.1. A Langevin (Brownian) dynamics simulation. The binding (unbinding) of Ca++ ions to (from) a fixed substrate S (e.g., CaM, AM, etc.) can be described in a Langevin simulation at various degrees of molecular resolution. The simplest way is to describe the binding sites as appropriately calibrated potential wells and count the number of occupied wells as a function of time. A trajectory that hits a free pump on the boundary of the spine head or reaches the dendritic shaft at the bottom of the spine neck is terminated there. The remaining part of the boundary is reflecting to trajectories. This is essentially the simplified molecular dynamics simulation described above. A coarse-grained description of the reaction of binding and unbinding of the diffusing ions with the immobile substrate is given by the reaction-diffusion equations (4.6)–(4.10). A numerical study of this system will be presented in a separate paper. The results of a full Langevin simulation based on our simplified physical and mathematical model are summarized below (see [26] for details). These results can be used as benchmarks for the results of the coarse-grained model described above. AM Figure 2 shows the results of a simulation with Ninit = 100, k−1 = Kback = cal 1 kHz, k1 = Kback = 5 Hz/M, R = 0.5µm, l = 0.2µm, d/2 = µm, Npumps = 10, and the protein molecules (about 50 of AM type and 10 of the other type) are distributed in the PSD. Two types of decay can be discerned in the first graph of calcium concentration vs. time in Figure 2: quick decay, starting at the beginning of the simulation and ending at about 250 msec, is followed by a slower decay that continues to the end of


1

# Ca2+ ions

100

2

100

80

60

40

40

20

20

40

0.6

30

0.4

20

0.2

1

1 2 3 4

2

3

4

10

0

1

2

3

# saturated proteins

6

1

15

7

100

0

0

1

2

3

4

Time (100ms)

2

3

4

0.8

80

0.6

60

0.4

40

0.2

20

0

0

0

1

2

3

4

0

1

2

3

8

6

10

5

1

Time (100ms)

# active pumps

5

0

4

Time (100ms) 20

4

50

0.8

80

60

3

1

# bound proteins

a)

15

4

5 4 3 2 1 0

0

1

2

3

4

Time(100ms)

Fig. 2. Dynamics of 100 calcium ions in dendritic spine. Time evolution of the concentration and binding. First row, concentration vs. time (in µsec), from left to right: 1. [Ca2+ ] in the total spine. 2. [Ca2+ ] in spine head. 3. Number of ions in the neck. Note that the neck contains only one ion at a time. 4. Number of bound proteins (type 1, blue, type 2, green). Note the stochastic nature of those curves. Second row, from left to right: 5. Number of saturated proteins of type 1 vs. time. 6. Arrival times of ions at active pumps: the ions leave one at a time. 7. Number of bound ions vs. time. 8. Number of active pumps vs. time. (Figure reprinted from [D. Holcman, Z. Schuss, and E. Korkotion, Calcium dynamics in dendritic spines and spine motility, Biophysical Journal, 87 (2004), pp. 81–91.] with permission.

the simulation. The first period is the decay curve of the saturation of type 1 proteins. When a simulation starts with 100 ions, only 10 proteins get saturated; that is, only about 40 ions are captured at the beginning and the number of saturated proteins continues to decay in time. To have a rough idea of the effect of the hydrodynamical push, we can approximate the push by its average of 2.5 proteins saturated for the first 250 msec, where each protein contributes to the speed a total of 50 nm/msec. The total speed of the push is 0.5 µm/ms. The push speeds up the arrival of ions at the lower part of the spine head, where the pumps are located, relative to arrivals by pure diffusion. Since the sojourn time of ions in the pumps is chosen to be short, the ions leave mainly through the head. At 500 msec into the simulation only about 20% of initial ions are still in the spine. In this simulation the effect of the push is not sufficiently strong to direct all the ions toward the neck. The 1:4 ratio of the efflux through the pumps, compared to that through the dendrite, may be due to the large number of fast pumps. These results are in agreement with the experimental results of [19]. In Figure 3 the results of simulations with and without the hydrodynamical push are shown: blue curves correspond to a simulation without the push effect, while magenta curves correspond to simulations with it. The characteristic parameters of AM cal the simulation are Ninit = 200, k−1 = Kback = 10 kHz, k1 = Kback = 0.5 kHz/M, R = 1 µm, l = 0.3 µm, and d/2 = 0.3 µm. There are 4 pumps, 60 AM proteins, and 10 calcineurin proteins [15]. Results similar to those predicted in Figure 3 were

16


200 400 600 800 1000 1200 1400 1600 1800 2000 500

1000

1500

2000

2500

Fig. 3. Comparison of the time evolution for a postsynaptic distribution with and without push. Blue curves correspond to a simulation without the push effect, while magenta curves correspond to simulations with it. First row, concentration vs. time (in µsec). From left to right: 1. [Ca2+ ] in the total spine. 2. [Ca2+ ] in spine head. 3. Number of ions in the neck. Note that the neck contains few ions at a time. 4. Number of bound proteins (type 1, blue, type 2, magenta). Note the stochastic nature of those curves. Second row, from left to right: 5. Number of saturated proteins of type 1 vs. time. 6. Arrival times of ions at active pumps: the ions leave one at a time. 7. Number of bound ions vs. time. 8. Number of active pumps vs. time. (Figure reprinted from [D. Holcman, Z. Schuss, and E. Korkotion, Calcium dynamics in dendritic spines and spine motility, Biophysical Journal, 87 (2004), pp. 81–91.] with permission.

obtained recently with calcium replaced by an inert dye that does not bind to proteins [27]. 7. An estimate of a decay rate. In the absence of the flow field V (x, t), the decay of M (x, t) is governed by the first eigenvalue of the Laplace operator in the head with the mixed reflecting and absorbing boundary conditions. In the presence of V (x, t), the decay rate can be estimated as follows (see also [37] for another estimate of the fast rate constant, using internal buffer kinetics). Consider the dynamics √ x˙ = v(t) + 2D w˙ x , y˙ =

√

2D w˙ y ,

z˙ =

√

2D w˙ z

in the neck, where wx , wy , and wz are independent Brownian motions. The solution is t √ x(t) = x0 + v(s) ds + 2D wx (t), 0

y = y0 +

√

2D wy (t),

z = z0 +

√

2D wz (t).

This means that the solution to the Nernst–Planck equation (4.4) in the neck, ct = D∆c − v(t)cx ,


17

is given by

G(y, z, t | y0 , z0 ) neck  2  t      v(s) ds  x − x0 −   0 f (x0 , y0 , z0 ) dx0 dy0 dz0 , × exp −   4Dt       c(x, y, z, t) =

where f (x0 , y0 , z0 ) is the initial ionic density in an infinite neck and G(y, z, t | y0 , z0 ) is Green’s function for the diffusion equation in the cross-section of the neck, with reflecting boundary conditions. We consider the initial decay law, when the decay is due primarily to the hydrodynamical effect, because it is faster than that due to diffusion. Suppose that the ions are concentrated near a single point (x0 , y0 , z0 ); that is, f (x0 , y0 , z0 ) = δ(x0 − x0 , y0 − y0 , z0 − z0 ). In the initial tenths of a second, the decay of the concentration in the neck is due to the large hydrodynamical effect. The velocity v(t) is maximal when all proteins are saturated. We write  2  t      v(s) ds  x − x0 −   0 c(x, y, z, t) = G(y, z, t | y0 , z0 ) exp −   2Dt       and x − x0 − (7.1)

2

t

0

v(s) ds =

2Dt

(x − x0 )2 (x − x0 ) v¯2 (t) − v¯(t) + t, 2Dt D 2D

where v¯(t) =

1 t

0

t

v(s) ds ≈ const ≡ v¯0 .

Set (x − x0 )2 ud (x, y, z, t) = G(y, z, t | y0 , z0 ) exp − 4Dt to represent the diffusion term. We can write in (7.1) −

(x − x0 )¯ v (t) v0 (x − x0 )¯ ≈− , 2D 2D

so this term does not contribute to the time decay of the concentration c(x, y, z, t) in v ¯02 the initial period. The last term in the exponent (7.1) is approximately 4D t, so in the limit of fast binding, which lasts a few hundreds of milliseconds, we can write v¯02 t . c(x, y, z, t) = Cud (x, y, z, t) exp − 4D

18


This gives the decay time (7.2)

τ=

4D . v¯02

The initial average velocity v¯0 can be estimated, if we assume that all proteins are distributed along the surface of the head and are saturated at the same time. In this case the membrane shrinks on the time scale of a single protein contraction time and of length equal to the number of proteins, Np = Ω S0 (x) dx, times the contraction length of a single protein. We consider two models of saturation. First, if the proteins are located on the membrane, we can say that the lengths do not sum, but act in parallel to contract the head. This yields a contraction length of order lc , the length of contraction of one protein, independently of the number of proteins distributed on the surface. Knowing the size of the myosin protein and the size of the head (e.g., radius of 1 µm), the maximal number of proteins packed on the membrane surface is 1 mM(= 600 proteins). Second, if there are different layers of contractile proteins, then all contractions add together, if they occur simultaneously. In that case the length of the contraction equals Np lc . We have (see Appendix)  4πR02 V¯0   in dimension 3,    |ΣN | v¯0 = F¯ (V (0)) =   2πR0 V¯0   in dimension 2,  |ΣN | where V¯0 = vQ Np is the average initial velocity of the surface of the spine head, as given by (4.17) with S (4) (0) = Np (by assumption); R0 is the initial radius of the spine head; and |ΣN | is the surface area of the cross-section of the neck. The value vQ = 0.02 µm/ms is given in [2], so that v¯0 = 0.1 µm/ms; hence  0.4πR02    in dimension 3,   |ΣN | ¯ v¯0 = F (V (0)) =   0.2πR0   in dimension 2.  |ΣN | Now, (7.2) gives τ = 160 msec, which is comparable to the experimental result given in [19]. This result can be obtained also from the following calculations. First, using the assumption that each protein contributes additively to the total contraction in the simulation of [26], we see that during the hydrodynamical push period about 5 contractile proteins are saturated on the average, so the average velocity of the head is V¯0 = 0.02 × 5 = 0.1 µ/ms. Second, if in our model all proteins are distributed in a single layer and are instantaneously saturated, they produce a contraction of 0.02 µm/sec. In this case, when the ratio of surface areas of the head and the neck (|ΣH | + |ΣN |)/|ΣN | = 5, the average velocity of the head is v¯0 . 8. Discussion and conclusions. We have introduced a Brownian dynamics simulation of calcium dynamics in a dendritic spine and its coarse-grained continuum


19

description by partial differential equations. The equations couple the hydrodynamical flow, caused by spine motility, to the chemical reaction between the diffusing calcium and the immobile substrate. We have identified the dominant molecular mechanism of the fast macroscopic twitching as the contraction of the calcium saturated AM proteins. This contraction produces a hydrodynamical flow, which causes the fast decay of calcium. The decay rate has been derived theoretically, and the reactiondiffusion equations described here are the coarse-grained version of the Langevin simulation of [26]. They provide a mathematical description of the molecular events during the fast motility of the spine. The main goal of the coarse-graining is to capture the features of calcium dynamics with a small number of equations, ideally with a single ordinary differential equation, so that a comprehensive model of calcium dynamics in a spiny dendrite can be derived and the effect of hundreds of thousands of spines can be integrated and coupled to an action potential. In this context the dynamics of calcium can possibly be linked to the induction of synaptic plasticity. An important feature of the Langevin simulation is that the number of bonds per protein, as a function of time, can be followed and compared to the initial calcium concentration. The coarse-grained reaction-diffusion description of the Langevin simulation should be helpful in relating the threshold of initiation of synaptic plasticity, such as LTP, to the initial calcium concentration. One of the most significant results of this paper is the derivation of the decay rate from the fast motility of the spine. This result should be compared to the calcium extrusion rate in spines, as presented in [19]. The two very distinct decay rates suggest that the fast extrusion period can also be due to the spine fast motility. This phenomenon was ascribed in [19], [37] to the fast pumping of calcium ions into stores. Indeed, in the present paper we have neglected calcium stores and high concentration of buffers, which may impose a decay rate faster than diffusion. An improved model that includes a large number of buffers should reveal the precise contribution of buffers to the calcium fast decay rate, as compared to the rate imposed by the spine contraction. Such a model has to be based on the mechanism of interactions between diffusing calcium and the buffers. We conclude, on the basis of the present model and of the Langevin simulation of [26], that one of the possible roles in calcium dynamics of the spine’s fast motility is to increase significantly the fraction of ions that are directed toward the dendrite and the organelles, compared to the ions that are pumped out. 9. Appendix: The velocities V (t) and F (V (t)). In this section, we compute the velocity V (t) of the spine head surface, given by (4.17), and the velocity F (V (t)), used in the boundary condition (4.14). While V (t) is given in (4.17), the velocity in the neck, F (V (t)), has to be calculated. To calculate the cytoplasmic fluid velocity F (V (t)) at the surface of the sphere inside the neck, we make the following simplifying assumptions that lead to an explicit expression for the velocity of the efflux. The entire fluid displaced by the contraction of the spine head flows into the spine neck with a uniform velocity v(t) in a direction normal to the sphere. We also assume that the neck is sufficiently narrow so that all normals to the spherical surface inside the neck, ΣN , are parallel to the axis of the neck. Under these assumptions the volume displaced per unit time is 4πR2 (t)V (t) in dimension 3 and 2πR(t)V (t), where R(t) is the instantaneous radius of the head and

20


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