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we present the first extension due to the inclusion of both ends, and in section we study the two versions of the model for the hydrolysis within the filament. In the last section |
, we examine transient properties of a single filament using numerical simulations and we show that for these transient properties, the vectorial and random models lead to distinct behaviors. This suggests experiments that would allow to discriminate between the two models. |
Vectorial model of hydrolysis with activity at both ends ATP hydrolysis is a two steps process: the first step is the ATP cleavage which produces ADP-Pi, and which is rapid. The second step is the release of the phosphate (Pi), which leads to ADP-actin and which is by comparison much slower 19 ADP-Pi-actin and ATP-acti... |
11 as well as in the present work. In other words, what is meant by hydrolysis in all these references is the step of Pi release. In this section, we assume that this release of Pi is a vectorial process described as a single reaction with rate [MATH] |
Let us recall the main features of the phase diagram of our previous model which assumes that only one end is growing. The model has three different phases: two phases of unbounded growth and one phase of bounded growth. In one phase of unbounded growth (phase III), both the cap and the bulk of the filament are unbound... |
13 14 We now extend the single-end model by including dynamics at both the ends. We keep, as before, the assumption of vectorial hydrolysis, which means that there is a single interface between the ATP subunit and ADP subunits, and the assumption of a reservoir of free ATP subunits in contact with the filament. The add... |
In Fig. we have pictorially depicted all these moves discussed above. Furthermore, we have assumed that all the rates are independent of the concentration of free ATP subunits [MATH] except for the on-rate which is [MATH] . All the rates of this model have been determined precisely experimentally except for [MATH] . Th... |
The state of the filament can be represented in terms of [MATH] the number of ADP subunits and [MATH] the number of ATP subunits. The dynamics of the filament can be mapped onto that of a random walker in the upper-quarter plane [MATH] |
with the specific moves as shown in figure We find the following steady-state phases (see Appendix A for details): a phase of bounded growth (phase I), and three phases of unbounded growth (phase IIA and IIB, phase III). The phase of bounded growth (phases I) and the phase of unbounded growth with unbounded cap (phase ... |
[MATH] but there is a section of ADP subunits which remains constant in time near the pointed end (this is analogous to the cap of ATP subunits near the barbed end in phase IIA). |
This phase diagram can be understood from the random walk representation of figure The velocity of the random walker in the bulk has components [MATH] along the [MATH] axis and [MATH] along the [MATH] axis, where [MATH] is the subunit size. Depending on the signs of these quantities, four cases emerge. If [MATH] and [M... |
[EQUATION] is finite, and the average filament velocity is (see Appendix A) [EQUATION] At the critical concentration [MATH] [MATH] and this marks the boundary to phase I. We find that |
[EQUATION] which is always larger than the critical concentration of the barbed end alone. In region III, the velocity is still given by |
[EQUATION] Similarly, in phase IIB, the probability of finding a non-zero region of D-subunits [MATH] is finite, and the average filament velocity is |
[EQUATION] which vanishes when [MATH] at the boundary with phase I, with [EQUATION] Note that [MATH] does not enter in [MATH] since the hydrolyzed part of the filament is always infinitely large in this case, in contrast to the case of |
[MATH] , which depends on both [MATH] and [MATH] . Note also that the velocity [MATH] and [MATH] are sums of a contribution due to the barbed end and a contribution due to the pointed end. This is due to the fact that in all growing phases, the filament is infinitely long in the steady state, and therefore the dynamics... |
Length fluctuations of the filament are characterized by a diffusion coefficient which is defined in Appendix A. Since the dynamics of each end is independent in phase IIA, the diffusion coefficient of this phase [MATH] is the sum of a contribution from the barbed end and another from the pointed end. From Ref. 11 we o... |
[EQUATION] where [MATH] is contribution of the diffusion coefficient due to the pointed end. On the boundary lines [MATH] and [MATH] , the average filament velocity vanishes. At this point, the addition of subunits at the barbed end exactly compensates the loss of subunits at the pointed end. Such a state is well known... |
21 . There, the length diverges as [MATH] near [MATH] and similarly as [MATH] near [MATH] as shown in figure , where [MATH] and [MATH] are diffusion coefficient in phases [MATH] and [MATH] . That divergence is a consequence of the assumption that the filament is in contact with a reservoir of monomers, in experimental ... |
As mentioned above, since the two ends are far from each other in the growing phases, they can be treated independently. In the phase of bounded growth (phase I) however, where the filament length reaches zero occasionally, the two ends are interacting strongly. For this reason, a precise description of the phase of bo... |
Hydrolysis within the filament: a vectorial or random process ? 2.1 Growth velocity As explained earlier, we have used a simplified model for hydrolysis 12 , in which the first step of hydrolysis is absent. The only remaining step, the phosphate release, is assumed to be a vectorial process. In the following, we keep t... |
We have compared experimental data from 22 together with the two theoretical models, vectorial and random. Both models successfully account for the observed sharp bend in the velocity versus concentration plots observed near the critical concentration as shown in figure . Below the critical concentration, the velocity ... |
Note that the velocities of both models superimpose, which means that bulk velocity measurements do not allow to discriminate between these models. Irrespective of the actual hydrolysis (phosphate release) mechanism, a fit of this data provides a bound on the value of the hydrolysis rate in the vectorial model [MATH] w... |
In figure , the phase diagram of the random hydrolysis model is shown. This phase diagram has only two phases in contrast to the vectorial case, because it can be shown that the average of the total amount of ATP subunits [MATH] is always bounded in the random model. Thus phase III is absent in the random model. In app... |
and which agrees well with the Monte Carlo simulations. 2.2 Length diffusivity Length fluctuations are quantified by the length diffusivity also called diffusion coefficient [MATH] which is defined in Eq. 18 of Appendix A. The length diffusivity of single filaments has been measured using TIRF microscopy by two groups ... |
Several studies have been carried out to explain this discrepancy: Vavylonis et al. computed the diffusion coefficient [MATH] as a function of ATP monomer concentration and found that [MATH] is peaked just below the critical concentration and its maximum is comparable to the value observed in experiments( [MATH] monome... |
According to Stukalin et al 12 and Vavylonis et al , the large length diffusivity observed in experiments results from dynamic instability-like fluctuations of the cap. It is important to point out that both papers make very different assumptions: the first one describes hydrolysis as a single step corresponding to Pi ... |
We have shown in figure the concentration dependance of [MATH] for the vectorial model using analytical expressions provided in the appendix and similar to that of 12 11 In this figure, the critical concentration defined as the boundary between phases I and II almost coincides with the concentration at the boundary bet... |
In figure , we have compared these analytical results obtained for the vectorial model with numerical results obtained for the random model. In the random model, we use Monte Carlo simulations to calculate a time dependent diffusion coefficient |
[MATH] , defined as [MATH] . For concentrations larger than the critical concentration, the initial condition is [MATH] , whereas for concentrations smaller than the critical concentration, the initial condition was a very long filament ( [MATH] |
subunits) with all subunits in the hydrolyzed state. On a large time window, we find that [MATH] is approximately time independent, and we interpret that value as the length diffusivity of the random model. Our results fully agree with that of Ref. , and with that of Ref. 20 |
in the limit of zero fragmentation rate. The length diffusivity indeed reaches a maximum of the order of 30 monomer /s below the critical concentration. As shown in that figure, there is only a small difference of length diffusivity in the vectorial case as compared to the random case: the maximum of the curve for the ... |
To make further progress, it would be very useful to reproduce experiments similar to those of Ref. 27 , on single filaments for various monomer concentrations, to confirm the scenario presented above for the length fluctuations of actin. Given that the predictions of the random and vectorial model are rather close to ... |
Dynamics of the filament in transient regimes Since it appears difficult to distinguish the vectorial from the random model using measurements of growth velocity or length diffusivity, one can turn to an analysis of the dynamics of the filament length in polymerization 27 |
or in depolymerization 10 to discriminate between the two models. Here, we focus on the dynamics of polymerization of a single filament, in the presence of a constraint of conservation of the total number of subunits (free+polymerized). This constraint leads to a steady state with a constant average length for the fila... |
In Fig. we show the filament length as well as its variance, as a function of time, for both vectorial and random models. Using Monte Carlo simulations we computed [MATH] , starting from [MATH] , for 1000 different realizations and calculated |
[MATH] Concerning the lag time of hydrolysis, we have observed that in simulations of the vectorial model, the filament typically reaches its steady state length long before it has been completely hydrolyzed. The time when this happens corresponds to the point where the two curves meet in Fig. (a). This characteristic ... |
[MATH] [MATH] min (with [MATH] , which is much longer than the typical time to reach the steady state [MATH] s. In contrast to this, in the random model, the time for completion of hydrolysis is comparable to the time to reach steady state (see Fig. (b)) as both the filament and the ADP part have similar growth dynamic... |
In practice, this lag time of hydrolysis may be difficult to measure on single filaments since the ATP subunits and ADP subunits can not be distinguished easily experimentally. In view of the previous section, on the role of ATP hydrolysis in length diffusivity, we suggest to study instead the length fluctuations of th... |
Thus contrary to velocity and length diffusivity measurements, an analysis of either the lag time of hydrolysis or of the time dependence of the length fluctuations provide a direct signature of the underlying mechanism of hydrolysis. |
Conclusion In this article, we have analyzed several aspects of the dynamics of a single actin filament. Many results discussed above could be extended |
mutatis mutandis to the case of microtubules. We have constructed a phase diagram, which summarizes all the possible dynamical phases of an actin filament with two active ends and vectorial hydrolysis in its inside. We have found that quantities like the filament velocity and the length diffusivity show similar behavio... |
We hope that our study will contribute to the understanding of the non-equilibrium self-assembly of actin/microtubule filaments. |
4.1 Acknowledgements We thank M. F. Carlier, J. Baudry, I. Fujiwara and A. Kolomeisky for illuminating discussions. We also acknowledge discussions with S. Sumedha, B. Chakraborty and F. Perez. We thank D. Blair for his contribution to the numerical study of the random model. We acknowledge support from the Indo-French... |
Appendix Appendix A Equations of the vectorial model with two ends Let [MATH] be the probability of having [MATH] hydrolyzed ADP subunits and [MATH] unhydrolyzed ATP subunits at time [MATH] , such that [MATH] is the total length of the filament. It obeys the following master equation: For [MATH] and [MATH] we have |
[EQUATION] For [MATH] and [MATH] [EQUATION] For [MATH] and [MATH] we have, [EQUATION] If [MATH] and [MATH] , we have [EQUATION] We define the following generating functions |
[EQUATION] Normalization imposes that at all times [MATH] [EQUATION] Using eqs 10 and 11 , we obtain the evolution equation for [MATH] |
[EQUATION] From [MATH] , the following quantities can be obtained: the velocity of the filament, which is [EQUATION] the diffusion coefficient characterizing filament length fluctuations |
[EQUATION] The average cap velocity is [EQUATION] and the diffusion coefficient characterizing the fluctuations of the cap is [EQUATION] |
Phase diagram and average length in the bounded phase To construct the phase diagram, we first focus on steady-states solutions of Eq. 16 , which are such that [MATH] The obtained equation for [MATH] involves the following time independent quantities |
[EQUATION] which are coupled back to [MATH] Progress can be made by considering two particular cases for [MATH] and [MATH] of this expression for [MATH] . This leads to |
[EQUATION] These two equations involve three unknowns [MATH] : the probability that the cap is zero, [MATH] : the probability that the [MATH] part of the filament is zero, and [MATH] : the probability that the filament is in the state of monomers. Note that [MATH] in phases of unbounded growth whereas [MATH] in the pha... |
In the random walk representation of figure the velocity of the random walker in the bulk has components [MATH] along the [MATH] axis and [MATH] along the [MATH] axis. Depending on the signs of these quantities, four cases emerge. If [MATH] and [MATH] , both the filament and cap length increase without bound (phase III... |
If [MATH] and [MATH] , the cap length remains constant in time which means [MATH] , but the rest of the filament made of D subunits can be either bounded (for [MATH] , which corresponds to phase I) or unbounded (for [MATH] which corresponds to phase IIA). When reporting the condition [MATH] into Eqs. 24 25 and solving ... |
An alternative way to find this condition is to start from the time dependent evolution equation of [MATH] of Eq. 16 and impose [MATH] . We end up with two coupled dynamical equations for [MATH] and [MATH] . The way to obtain the velocity and diffusion coefficient in phase IIA from these equations is explained in detai... |
Similarly, if [MATH] and [MATH] , the length of the region of D subunits at the pointed end remains constant in time, and the region of T subunits can be either bounded (phase I) or unbounded (phase IIB). By either method, one obtains the velocity in the phase IIB given in Eq. , and the condition that marks the boundar... |
In Ref. 11 , an explicit expression for the average length in the phase of bounded growth was obtained by a method of cancellation of poles of [MATH] . Unfortunately, this method does not allow us to derive the expression of [MATH] here, because the rates [MATH] and [MATH] lead to an additional unknown [MATH] in Eq. 16... |
Appendix B Mean-field equations of the random model We explain in this appendix how the velocity of the filament in the random model is obtained from a mean-field approach. This appendix is provided mainly for pedagogical reasons, since the solution has already appeared in Ref. 12 and Ref. 26 For simplicity, we focus o... |
[MATH] inside the filament an occupation number [MATH] , such that [MATH] if the subunit binds ATP and [MATH] otherwise. In the reference frame associated with the end of the filament, the equations for the average occupation number are |
[EQUATION] [EQUATION] In a mean-field approach, the effect of correlations [MATH] are neglected, i.e. these correlations are replaced by [MATH] |
(and similarly for averages of product of three occupation numbers). At steady state, the left-hand sides of Eqs. 26 27 are both zero, which leads to recursion relations for the [MATH] . Note that [MATH] is denoted as [MATH] in Ref. 26 and as [MATH] in Ref. 12 We still denote [MATH] , since it represents the probabilit... |
[EQUATION] Combining Eqs. 26 28 , one obtains the following cubic equation for [MATH] [EQUATION] This cubic equation has three solutions, but only one solution is such that [MATH] . The rate of elongation of the filament can be obtained by reporting that solution into |
[EQUATION] In figure , this velocity [MATH] is shown as function of the concentration of free monomers. For low values of [MATH] , the velocity of the random and vectorial model are identical, as [MATH] is increased the velocity of the random model starts to deviate from the curve of the vectorial model. By imposing th... |
Figure Legends Figure Schematic diagram representing the addition of subunits with rate [MATH] , removal with rates [MATH] [MATH] and [MATH] , and hydrolysis with rate [MATH] , which can only occur at the interface between T and D monomers in the vectorial model. Note that two new rates [MATH] and [MATH] have been adde... |
Figure Representation of the various possible moves for actin dynamics. (i), (ii) and (ii) depict different cases for vectorial hydrolysis. (iv) and (v) depict cases for random hydrolysis. |
Figure Theoretical phase diagram for the vectorial model with two ends in the variables hydrolysis rate [MATH] and on-rate [MATH] . The line OQ is obtained by setting the cap velocity equal to zero, and the line OP is given by the condition [MATH] where [MATH] is the velocity in phase IIA calculated in Eq. . Similarly,... |
is the velocity in phase IIB given in Eq. Figure Filament velocity [MATH] versus concentration of free monomers [MATH] for the vectorial model with two active ends. (a) Case [MATH] for [MATH] . In regions I and IIA, |
[MATH] , where [MATH] is given by Eq. . In region III, [MATH] , where the velocity is that of Eq. (b) Case [MATH] for [MATH] . Here [MATH] where [MATH] is given by Eq. |
Figure Average length as function of concentration. (filled circles ) [MATH] and [MATH] ; (open circles) [MATH] and [MATH] ; (open squares) [MATH] |
and [MATH] ; (filled squares) [MATH] and [MATH] . The rates which are not specified here are given in table . The black line is [MATH] |
Figure Phase diagram of the random hydrolysis in the coordinate on-rate [MATH] versus hydrolysis rate [MATH] (per site). The symbols have been obtained from Monte Carlo simulations, while the solid line is the mean-field theory of appendix B. For [MATH] , we recover the value of [MATH] corresponding to the critical con... |
Figure Velocity versus free monomer concentration. The squares symbols are experimental data of 22 , which were taken from Ref. 32 , the solid lines is the velocity for the random model as calculated from the theory presented in appendix B and the plus symbols is the velocity for the vectorial model using rates in tabl... |
Figure Diffusion coefficient as function of the monomer concentration for the random and vectorial model of hydrolysis. The data points are the prediction for the random model of hydrolysis while the solid lines are the predictions for the vectorial model. The dashed (resp. dash-dotted) vertical line represents the cri... |
Figure (a) and (b) : Total filament length (denoted [MATH] , black), and total amount of hydrolyzed subunits (denoted [MATH] , grey) as function of time for the case of vectorial hydrolysis (left panel) and random hydrolysis (right panel) (the total concentration of subunits |
[MATH] ; 1 filament in a volume of 10 [MATH] ). Note that the point where the two curves meet in the random hydrolysis model occurs much earlier compared to the case of vectorial hydrolysis ( [MATH] 10000s). (c) and (d): The variance [MATH] ) as a function of time is plotted for the vectorial model and random model res... |
# Source: arxiv 0908.2827 # Title: A model for the generation of localized transient Na+ elevations in vascular smooth muscle # Sections: all # Downloaded: 2026-03-03T05:15:01.909813+00:00 |
A model for the generation of localized transient [MATH] elevations in vascular smooth muscle Nicola Fameli, Kuo-Hsing Kuo, Cornelis van Breemen |
Department of Anesthesiology, Pharmacology and Therapeutics The University of British Columbia Abstract We present a stochastic computational model to study the mechanism of signalling between a source and a target ionic transporter, both localized on the plasma membrane (PM) and in intracellular nanometre-scale subpla... |
[MATH] waves in vascular smooth muscle (VSM), the physical and functional link between non-selective cation channels (NSCC) and [MATH] [MATH] exchangers (NCX) needs to be elucidated in view of two recent findings: the identification of the transient receptor potential canonical channel 6 (TRPC6) as a crucial NSCC in VS... |
refilling, this quantitative approach now allows us to model the upstream linkage of NSCC and NCX. We have implemented a random walk (RW) Monte Carlo (MC) model with simulations mimicking [MATH] diffusion process originating at the NSCC within PM-SR junctions. The model calculates the average [MATH] in the nanospace an... |
[MATH] source. Our results highlight the necessity of a strategic juxtaposition of the relevant signalling channels as well as other physical structures within the nanospaces to permit adequate [MATH] build-up to provoke NCX reversal and |
[MATH] influx to refill the SR. Keywords: calcium oscillations, calcium signaling, sodium transient, vascular smooth muscle, sarcoplasmic reticulum, stochastic model, monte carlo, random walk, computational model, TRPC6, [MATH] [MATH] exchanger. |
Introduction We are not used to thinking of ionic sodium ( [MATH] as a signalling ion, despite its known relevance for vascular disease. On the other hand, the importance of the second messenger [MATH] in signalling cell function is undisputed. There is however recent experimental evidence supporting the idea that this... |
[MATH] particularly in signalling events that are allowed by the juxtaposition of signalplexes and transporter carrying membranes at nanometric distances from one another. This observation, and others outlined below, as well as a wealth of accumulated knowledge on [MATH] and |
[MATH] transporters in VSMC has prompted us to take a more quantitative look at the question of [MATH] -related signalling in PM-SR cytoplasmic nanospaces (also referred to as nanodomains or junctions): nanometre-scale signalling compartments comprising the PM, the sarcoplasmic reticulum (SR), [MATH] [MATH] and other i... |
In this article we present a quantitative model aimed at elucidating the mechanism of selective (or site-specific) signalling between a source ionic transporter and a target ionic transporter, both of which are localized on the same membrane and are part of a nanodomain. We developed this model using a stochastic metho... |
[MATH] gradient, which, in turn, enables reversal of a NCX (presumably in the vicinity of the NSCC) and consequent NCX-mediated [MATH] entry into the subplasmalemmal nanodomain. A few examples highlighting the biological importance of this intra-membrane system and its link to pathogenic mechanisms are (table |
summarizes these cases): 1) A critical subplasmalemmal step in the [MATH] signalling cascade giving rise to VSM cell contraction following G-protein coupled receptor adrenergic stimulation |
—where blocking NCX reversal causes attenuation and elimination of [MATH] oscillations due to impairment of SR refilling; replacement of [MATH] oscillations with a tonic [MATH] signal causes a dramatic decrease in smooth muscle force development |
in pulmonary artery SM, upregulation of the NCX and [MATH] entry via reverse NCX action is considered one of the mechanisms behind elevated |
[MATH] in idiopathic pulmonary arterial hypertension patients 2) Receptor activated [MATH] entry may also induce NCX reversal in endothelial cells (EC) and this, in turn, gives rise to selective [MATH] -stimulated eNOS activity and NO production; eNOS derived NO is an important physiological vasodilator agent and is ac... |
; in EC NCX operating in forward mode is also important in regulating both [MATH] and [MATH] ; this suggests that local [MATH] besides the cell membrane potential and the equilibrium potential of the NCX, have a role in the regulation of forward NCX too |
3) In nerve terminals, a transient [MATH] increase linked to a tetanic pulse of the action potential can reverse the NCX to induce presynaptic potentiation; this observation implies a role for the NCX in synaptic facilitation and has consequences for short term memory from reverse NCX malfunction |
4) In skeletal muscle NCX plays an important role in [MATH] homeostasis, operating in forward as well as reverse mode in that function; |
[MATH] [MATH] exchange is involved in the control of muscle fatigue and there are reports supporting the notion that the beneficial role of external [MATH] in protecting slow-contracting soleus muscle against high-frequency fatigue depends mostly on [MATH] influx through reversal of the NCX |
5) although the role of NCX in heart is still poorly understood, both its [MATH] -efflux and influx modes are observed in cardiac myocytes and and the latter is likely a consequence of subplasmalemmal [MATH] elevations |
All of the above emphasize that modulation of [MATH] signaling via [MATH] and [MATH] [MATH] exchange is of great clinical relevance in areas such as hypertension, chronic heart failure and possibly cerebral and skeletal muscle malfunction. |
This study sheds new light on a few new scientifically interesting key factors regarding the “workings” of intracellular signalling nanospaces. We investigate the role, and importance, of having a confining surface, namely another lipid membrane, facing the membrane where the source and target transporters belong. This... |
but it would appear that it alone cannot be entirely responsible for the generation of a sufficiently large local [MATH] transient elevations. We find that to understand how sufficient [MATH] can be built up within the nanodomain to trigger NCX reversal and |
[MATH] signalling downstream, it is important to account for other factors such as the role of physical obstructions to [MATH] diffusion and the possible organization of these obstructions. |
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