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Methods 2.1 Electron microscopy Details of the electron microscopy have been described previously The primary fixative solution contained 1.5% glutaraldehyde, 1.5% paraformaldehyde and 1% tannic acid in 0.1 M sodium cacodylate buffer that was pre-warmed to the same temperature as the experimental buffer solution (37 C)... |
2.2 Simulations Simulations of [MATH] diffusive motion from a source transporter are based on an implementation of a Monte Carlo random walk (RW), along the lines of methodology previously employed for [MATH] transport simulations |
The model nanospace in which the simulations of [MATH] diffusion take place arises from observed physical features and properties of the essential elements for nanodomain signalling. These are either obtained from our laboratory’s studies or from the available literature |
and they are essentially the TRPC6 cytosolic ‘radius’ and [MATH] , the typical dimensions of intracellular nanospace ultrastructure, estimates on the number of pillars, [MATH] diffusivity in cytosol, expected [MATH] necessary for NCX reversal (see section |
3.1 and during localized [MATH] transients (LNats) observed in Fig. is a to-scale representation of the geometry of the model nanospace used in the simulations. |
In the simulations, particles representing [MATH] performs a RW on a cubic lattice with spacing [MATH] nm; initially picked as an approximation to the expected [MATH] mean free path in water (in turn, as an estimate to the mean free path in cytosol), we also carried out tests for effects of varying this parameter in a ... |
The RW time step [MATH] was chosen by running several simulations letting particles cover a predetermined straight line distance [MATH] , and recording the number of RW steps [MATH] taken to cover [MATH] From diffusion theory, the total time [MATH] |
taken by a random walker in three dimensions to cover the distance [MATH] is [MATH] , where [MATH] is the measured diffusivity of [MATH] in muscle cytosol |
. The quantity [MATH] was our choice for [MATH] and its value is [MATH] s. The boundary conditions in our simulations are as follows. At the PM and SR membranes we implemented reflecting conditions: ions arriving at one of those surfaces during their RW are reflected back into the nanospace. At the edges of the model n... |
Simulation code is written and tested in the C programming language on a computer running a Linux operating system. After testing and troubleshooting, programs are recompiled and run in one of the WestGrid computing nodes |
The pseudo-random number generator we used is the algorithm gsl_rng_m19937 of the GNU Scientific Library since it has sufficient randomness and quality requirements for our purposes. The plots with simulation results were produced using the “freely distributed plotting utility” gnuplot |
Results 3.1 Model foundation This laboratory has previously experimentally established that a [MATH] transient from a NSCC enabled the generation within a PM-SR junctional nanospace of a sufficiently high [MATH] |
gradient to cause NCX reversal A quantitative estimate of the level of such burst can be obtained by comparing the equilibrium potential of the NCX, [MATH] , with the membrane potential, [MATH] , using typical values for vascular smooth muscle cells. These quantities are linked by the equation [MATH] (where [MATH] betw... |
(In general, [MATH] , where [MATH] is the universal gas constant, [MATH] the absolute temperature, [MATH] is a [MATH] -valent cation, and |
[MATH] is Faraday’s constant.) NCX reversal occurs when [MATH] . We can study this inequality for both resting and activated cell conditions by a plot like the one in Fig. Using for an activated cell [MATH] mV, |
[MATH] mM, [MATH] [MATH] mM, [MATH] J/(mol K), [MATH] K, and [MATH] J/(V mol), we observe that, during activation, [MATH] transient of the order of 30 mM or greater is necessary to cause NCX reversal. In this exercise, we have used an estimated value for [MATH] , as that is approximately the lowest value [MATH] we expe... |
Now, equipped with (a) the fundamental observation of localized [MATH] elevation transients in full agreement with the values suggested by the study of Fig. |
(b) better knowledge of the identity of NSCC as TRPC6 (c) the basic idea that the presence of intracellular nanospaces is necessary for this signalplex to be complete, we propose a model to investigate the role of strategic placement of transporters with respect to each other, as well as of a confining membrane and oth... |
[MATH] to permit NCX reversal. The model nanospace used for the study is illustrated in Fig. . The dimensions expressed therein are based on high quality EM images showing that the PM-SR separation in these nanospaces is remarkably uniform and about 20 nm. Lateral extension of these closely apposed PM-SR regions is app... |
). 3.2 Random walk simulations: bare PM-SR nanospace The simplest model nanodomain we studied consists of a shallow cylinder-shaped volume of height [MATH] and radius [MATH] , as in Fig. [MATH] entering the nanodomain via an NSCC are represented as particles doing a RW based on a given diffusivity [MATH] corresponding ... |
For simplicity, there is only one [MATH] source positioned at the centre of the PM side of the nanospace (in section we will elaborate further on the issue of the number of [MATH] sources). The simulation programs output the computed [MATH] rise above resting level as a function of time, thereby giving an average [MATH... |
concentration in the nanodomain between PM and SR.) Results from this set of simulations are shown in Fig. , left panel. This plot helps establish the time scale after which we can consider that the [MATH] has reached a steady-state level. We can observe from the graph that after approximately 100 [MATH] the concentrat... |
3.3 Random walk simulations: randomly distributed obstacles in the nanospace Evidently, this simplest incarnation of the model is inadequate to describe the generation of [MATH] transients of the observed size |
since at steady state elevation of [MATH] hovers around [MATH] M or about three orders of magnitude less than the observed values of 15–20 mM. We need to consider other nanospace features emerging from our ultrastructural images that may be responsible for a larger increase in the [MATH] . Barring artificially changing... |
There is convincing evidence suggesting the existence of structures spanning the width of the nanospace and which could constitute an impediment to the free diffusion of [MATH] |
Our own observations confirm the existence of electron opaque “pillars” in transmission electron microscopy images like the one in Fig. . The size and abundance of these electron opaque structures compares well with the electron dense “bridges” observed by Devine and collaborators in the early ‘70s |
Keeping the overall geometry of the model nanospace the same (Fig. ), we have therefore implemented a number of junction spanning structures in the form of cylindrical pillars, having estimated their size from several images like the one in Fig. |
. From those same images it is possible to approximate the percentage junctional volume occupied by those structures and, in turn, an approximate number of them expected per nanospace. In a series of simulations, we have represented up to 200 pillars randomly distributed within the nanospace (Fig. ), and then simulated... |
In this case too, we let the simulations run for a time sufficiently long to ensure that a steady-state level for the [MATH] was established. Results are reported in Fig. In all simulations involving random positioning of pillars, to minimize bias from the particular random pillar distribution, we have take average val... |
The results of this series of simulations indicate that having some form of impediment to ionic motion in the junction does produce the effect of increasing the [MATH] and of changing its profile to one that decays more slowly with distance from the [MATH] source. We ensured that this effect was not simply a consequenc... |
[MATH] computed with different numbers of pillars in the junction (blue dots in Fig. ) and comparing it with an increase in [MATH] merely due to reducing the nanospace volume by the volume of the pillars (red line in Fig. ). The plot in Fig. |
demonstrates that ion collisions with pillars do indeed have a role in forcing [MATH] to dwell longer in the nanospace. 3.4 Random walk simulations: non-randomly distributed obstacles |
Clearly, the presence of obstacles to diffusion has an effect of increasing [MATH] , however the values we obtain this way are not yet comparable with those measured during the local [MATH] elevation transients. Other junctional features need to be accounted for in order to understand the mechanism giving rise to such ... |
We considered the hypothesis that nature might place these obstacles “strategically” rather than randomly, in a neighbourhood of a [MATH] |
source, so as to favour the generation of the gradients needed to drive the signalling chain. The rationale is that while a random set of pillars does show an ability to retain ions in the nanospace longer and therefore allow higher concentration build up, it does not do it efficiently enough to quantitatively account ... |
transients. We then ran some simulations in which pillars are placed in a circle around a [MATH] source in such a way that we can control the porosity of this pillar fence to the passage of random walking ions. (Imagining to stand where the [MATH] source is, the circle of pillars would appear as a 20-nm-high set of sla... |
Representative results from these simulations are reported in Fig. The dramatic increase in the [MATH] in the vicinity of the source caused by this type of obstacle configuration is immediately evident. |
[MATH] is in this case of the same order of magnitude as the observed localized [MATH] transient elevation phenomenon . Note furthermore that in this configuration the model suggests that the time for the [MATH] to reach steady-state is much longer than in the case of random distribution of pillars. The more scattered ... |
Discussion The stochastic computational model presented herein attempts to give a quantitative description of the mechanism behind the observed localized |
[MATH] transients observed in VSMC . Based as much as feasible on experimental observations of the physical and physiological features of the intracellular nanospaces in which these transients are hypothesized to occur, the model results lead us to conclude that |
[MATH] can build to sufficiently high values in PM-SR junctions to give rise to the observed transients. Three main steps lead to the fundamental hypothesis behind this work. The first stems from earlier work by this laboratory elucidating the sequence leading to asynchronous [MATH] oscillations in VSMC |
. Succinctly, a large external [MATH] influx causes the reversal of the NCX and consequent [MATH] entry to refill the SR after SR- [MATH] |
release via IP R channels upon cell stimulation provokes contraction. The second is the observation of two main features of the localized |
[MATH] elevation transients they appear as a punctate pattern on the periphery of VSMC (with puncta having a given time course), and their peak [MATH] |
values are comparable to the [MATH] values necessary to cause NCX reversal (see Fig. and relevant text). Thirdly, our previously published model supports the idea that NCX reversal-mediated [MATH] entry in PM-SR nanospaces introduces sufficient [MATH] to refill the SR during asynchronous [MATH] |
waves Based on these three points, the hypothesis we set out to study with our model is that the observed localized [MATH] transients occur in PM-SR nanodomains (or nanospaces or junctions), in other words, that in those junctions, due to [MATH] |
influx via a TRPC6 channel, [MATH] can attain levels, that cause [MATH] entry via NCX reversal. Simulation results from our simplest version of model (Fig. ), namely, a PM-SR nanospace filled with cytosol (represented in our simulations by the diffusivity of [MATH] with only one TRPC6 channel as |
[MATH] source suggest that this simple view is not adequate to describe the formation of [MATH] transients. This is mainly due to the large value of the diffusion coefficient of [MATH] which is about three times that of free [MATH] in cytosol, and the fact that there is no observed buffering effect of [MATH] in cytosol |
Recent [MATH] measurements in isolated rabbit ventricular myocytes by Despa and Bers ( also suggest that the effective diffusivity of [MATH] may be much slower than the one found by Kushmerick and Podolsky |
although no explanation as to the mechanism behind it was suggested. In among other things, Despa and Bers measured endogenous [MATH] buffering in cytosol and found it negligible when compared to that of other important species like [MATH] , for example. It is well known that [MATH] |
buffering lowers its diffusivity dramatically thus possibly contributing to easier generation of [MATH] gradients in confined spaces like the PM-SR nanospaces. Presuming the slight [MATH] buffering effect observed in cardiac cell cytosol is mirrored in smooth muscle cells, it clearly could not provide sufficient slowin... |
diffusivity to aid local transient creation. It seems instead ever more plausible that it must be some sort of physical obstruction to ionic motion that gives rise to a slower effective diffusivity for |
[MATH] We then considered that physical obstructions to ionic motion can have the overall effect of increasing the time spent by each [MATH] in the nanospace thereby increasing the value of [MATH] Ever since first reported measurement of [MATH] diffusivity in muscle cells |
, it was hinted that it is physical rather than chemical interactions that cause a retardation of the ionic (and non-ionic for that matter—sorbitol and sucrose specifically) intracellular diffusion. Our ultrastructural observations of nanospace spanning electron opaque structures support this idea (Fig. and agree fully... |
The introduction of such physical obstruction in the simulations seems then more than warranted and very plausible. Moreover, if the “tortuosity factor”, introduced by Kushmerick, arising from physical interaction is indeed important in the bulk cytosol, then it would be even more so in the restricted PM-SR nanospaces.... |
and ), to obtain a quantitative agreement with the observed level of [MATH] during the transients we need to hypothesize further that these junction spanning structures are not distributed at random, but rather form an organized barrier around a given [MATH] source within the nanospace. Our simulation results in this c... |
[MATH] with this configuration of pillars will reach steady state with a time constant that is at least ten times longer than in the case of randomly distributed pillars. This is another feature supporting our hypothesis that these transients are formed in the PM-SR nanospaces with the aid of a relatively tight fence o... |
It therefore appears that two features are essential if these transients must occur within PM-SR nanospaces: there must be physical obstructions to |
[MATH] motion, forming an organized barrier around each [MATH] source (TRPC6 channels) and NCX must be localized near a TRPC6 within such barrier to be able to sense the high [MATH] , reverse and allow [MATH] entry. The second of these two features is suggested by a number of observations. There are studies supporting ... |
Still other studies indicate that at least certain kinds of TRP channels are found in close proximity of caveolin . Combined with our earlier observations that NCX tend to crowd near the necks of caveolae |
this makes a strong case for a physical association between TRP channels and NCX. The first of the above mentioned desirable features—patterns of physical obstructions within the nanodomains—is a harder matter to study experimentally and one of the current/future directions our laboratory is following. |
To further study the issue of the localized [MATH] transient duration requires more computationally demanding simulations, which we are currently tackling. We can however make a qualitative argument to suggest how the time scale of the observed transients can also be supported by our model. We have so far only implemen... |
. If we conjecture that TRPC6 and NCX must be in physical proximity, as we speculated above, it is consistent to assume that there may be more than one TRPC6 in each junction functioning as a [MATH] entry gate, and perhaps even as many as there are NCX. |
Other transporters such as NCX and [MATH] /K ATPases (NKA) certainly also play a role in shaping the PM-SR nanodomain [MATH] profiles that drive NCX-mediated [MATH] entry during VSMC activation. In the simple model we presented in this article, we focussed on the role of TRPC6 since it is by far the larger capacity tra... |
Acknowledgments The research was supported by grants from the Canadian Institute of Health Research and the Heart and Stroke Foundation of British Columbia and Yukon. This project has been enabled by the use of WestGrid computing resources, which are funded in part by the Canada Foundation for Innovation, Alberta Innov... |
# Source: arxiv 0908.3377 # Title: A hybrid of the optimal velocity and the slow-to-start models and its ultradiscretization # Sections: all # Downloaded: 2026-03-02T08:58:16.172147+00:00 |
Also at ]Gunma National College of Technology, 580 Toriba, Maebashi, Gunma 371–8530, Japan A hybrid of the optimal velocity and the slow-to-start models and its ultradiscretization |
Abstract Through an extension of the ultradiscretization for the optimal velocity (OV) model, we introduce an ultradiscretizable traffic flow model, which is a hybrid of the OV and the slow-to-start (s2s) models. Its ultradiscrete limit gives a generalization of a special case of the ultradiscrete OV (uOV) model recent... |
optimal velocity (OV) model, slow-to-start (s2s) effect, ultradiscretization Introduction Studies on microscopic models for vehicle traffic provided a good point of view on the phase transition from free to congested traffic flow. Related self-driven many-particle systems have attracted considerable interests not only ... |
Whereas the OV model consists of ordinary differential equations (ODE), cellular automata (CA) such as the Nagel–Schreckenberg model Nagel1992 the elementary CA of Rule 184 (ECA184) Wolfram1986 , the Fukui–Ishibashi (FI) model Fukui1996 and the slow-to-start (s2s) model Takayasu1993 are extensively used in analyses of ... |
II The OV model and the s2s effect Imagine many cars running in one direction on a single-lane highway. Let [MATH] denote the position of the [MATH] -th car at time [MATH] . No overtaking is assumed so that [MATH] holds for arbitrary time [MATH] The time-evolution of the OV model Bando1995 is given by |
[EQUATION] where [MATH] and [MATH] are the velocity of the [MATH] -th car and the interval between the cars [MATH] and [MATH] , respectively. A function [MATH] and a constant [MATH] represent an optimal velocity and sensitivity of drivers, or the delay of drivers’ response, in other words. |
Since the current velocity and the current interval between the car ahead determine the acceleration through the time-evolution and the optimal velocity, we classify the OV model ( ) as the acceleration-control type (aOV). On the other hand, the OV model of the velocity-control type (vOV) was proposed in earlier studie... |
[EQUATION] Replacement of [MATH] in the above equation ( ) with [MATH] and the Taylor series of [MATH] yield [EQUATION] which is rewritten as |
[EQUATION] Thus we note that the aOV model ( ) is given by neglection of the higher derivatives in the Taylor series of the vOV model ( ). Though the aOV model is more common in the studies on vehicle traffic, we shall concentrate on an ultradiscretizable hybrid of the vOV and the s2s models. Thus we call the vOV model... |
Note that the input to the OV function [MATH] in the OV model ( is the headway at a single point of time [MATH] that is prior to the present time [MATH] . Thus we may say that the OV model describes, in a sense, “reckless” drivers since the model pays no attention to the headway between the time [MATH] and the present ... |
[MATH] containing information on the headway for a certain period of time going back from the present as an input to the OV function [MATH] We shall see this idea works in what follows. |
What is crucial in the ultradiscretization of the aOV model Takahashi2009 is the choice of the OV function, [EQUATION] where [MATH] and [MATH] are positive constants. In terms of the auxiliary functions, |
[EQUATION] the OV function ( ) is expressed as [EQUATION] A naive discretization of the auxiliary function ( ), [EQUATION] introduces the OV function for the discrete OV (dOV) model, |
[EQUATION] which is found to be ultradiscretizable. Takahashi2009 Let [MATH] and [MATH] where [MATH] and [MATH] are the integral time and the discrete time-step, respectively. Employing the effective distance as |
[EQUATION] where [MATH] we extend the OV model ( ) in a time-discretized form as [EQUATION] which is equivalent to [EQUATION] It is straightforward to confirm that the continuum limit [MATH] of the above discrete s2s–OV (ds2s–OV) model ( ) reduces to the integral-differential equation which we call the s2s–OV model, |
[EQUATION] where the corresponding effective distance is given by [EQUATION] We shall see that the s2s effect is indeed built into the OV model in the ultradiscrete limit of the ds2s–OV model. |
III Ultradiscretization Ultradiscretization Tokihiro1996 is a scheme for getting a piecewise-linear equation from a difference equation via the limit formula |
[EQUATION] In order to go forward to the ultradiscretization of the ds2s–OV model ( ), it will be a good choice for us to begin with the ultradiscrete limit [MATH] of the auxiliary function ( ): |
[EQUATION] In the same way to make the OV function for the dOV model ( from the auxiliary function ( ), we obtain the OV function for the uOV model Takahashi2009 as |
[EQUATION] where [MATH] The effective distance ( ), on the other hand, is ultradiscretized in the same manner: [EQUATION] Thus we obtain an ultradiscrete equation |
[EQUATION] which is equivalent to [EQUATION] as the ultradiscrete limit of the ds2s–OV model ( ). We name it the ultradiscrete s2s–OV (us2s–OV) model. When the monitoring period [MATH] is fixed at zero, the us2s–OV model reduces to a special case of the uOV model Takahashi2009 As we can see from eqs. ( 11 ), ( 12 and (... |
Now let us see how a CA comes out from the us2s–OV model. Let [MATH] be the discretization step of the headway [MATH] or equivalently, the size of the unit cell of the CA. Then with no loss of generality, we may set [MATH] Assume that the number of vacant cells between the cars [MATH] and [MATH] [MATH] must be non-nega... |
[EQUATION] Fixing [MATH] at an integer, we call this model the s2s–OV cellular automaton (CA). The s2s–OV CA reduces to the FI model Fukui1996 |
when [MATH] and to the ECA184 Wolfram1986 when [MATH] and [MATH] The s2s model Takayasu1993 also comes out from the s2s–OV CA by choosing [MATH] and [MATH] Thus the s2s–OV CA is regarded as a hybrid of the FI model and an extended s2s model. |
IV Numerical experiments We shall numerically investigate the s2s–OV CA ( 14 ). Throughout this section, the length of the circuit [MATH] is fixed at [MATH] and the periodic boundary condition is assumed as well so that [MATH] is identified with [MATH] |
Spatio-temporal patterns showing trajectories of each vehicle are given in fig. We choose the parameters and initial conditions so that jams appear in the trajectories. The two figures in the top share the same monitoring period |
[MATH] but their maximum velocities are different. The top left trajectories show that the velocities of the vehicles are zero or one, which is less than or equal to its maximum velocity |
[MATH] . In the top right trajectories whose maximum velocity [MATH] , on the other hand, the velocities of the vehicles read zero, one, two and three. Thus we notice that the vehicles driven by the s2s–OV CA can run at any allowed integral velocity which is less than or equal to its maximum velocity [MATH] |
The other two figures in the bottom in fig. share the same maximum velocity [MATH] , but their monitoring periods are different. As is observed in the bottom two figures, the longer the monitoring period is, the longer it takes for the cars to get out of the traffic jam. |
The jam front is observed to propagate against the stream of vehicles at constant velocity [MATH] , since cars have to wait [MATH] time-steps to restart after their preceding cars restarted, as is depicted in fig. |
Fig. shows fundamental diagrams giving the relation between the vehicle flow [EQUATION] which is equivalent to the total momentum of vehicles per unit length, and the vehicle density [MATH] , where |
[MATH] is the number of vehicles. The fundamental diagrams clearly show phase transitions from free to jam phases as well as metastable states, which are also observed in empirical flow-density relations Chowdhury2000 Helbing2001 |
It is remarkable that the fundamental diagrams have multiple metastable branches. This feature is similar to that reported by Nishinari et al. |
Nishinari2004 By observation, we note that each fundamental diagram has [MATH] metastable branches and a jamming line. The branches and the jamming line correspond to integral velocities that are less than or equal to the maximum velocity |
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