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Figure (6) exhibits the space-time plots of some more Kinesins and the time dependence of their centre of mass at [MATH] We observe that although each Kinesin undergoes rapid fluctuations upon becoming unbound, the centre of mass (CM) has a smooth increasing behaviour in time. We have found the CM velocity by fitting a... |
On the occasion of detachment from the MT, the Kinesin performs a stochastic random motion, in the cytoplasmic environment, which is characterized by large fluctuations. It is a well-known fact that Kinesins are processive motors i.e., they can move directedly along the filament for relatively large distances before de... |
As can be seen from Figure (7), the hand-over-hand mechanism of the directed motion is evident. In order to find a better insight, we have evaluated the dependence of a single Kinesisn mean squared displacement (MSD) on time. The MSD has been obtained by averaging over the trajectories of all the Kinesins. Figure (8) d... |
We have evaluated the slopes by linear curve fitting. To a very good approximation the diffusion is normal for densities larger than [MATH] i.e., the MSD grows linearly in time. It would be more appropriate to define a line density [MATH] . This quantity is defined as the time average number of bound Kinesins to the MT... |
We observe a linear increase of [MATH] up to [MATH] Afterwards the rate of increase is lowered. This is due strong repulsion between bound Kinesins. We stress that in order to have values of [MATH] larger than [MATH] we have to significantly increase [MATH] . This enormously rises the computational costs. We extrapolat... |
[EQUATION] In the extreme limit [MATH] we have (by extrapolation) [MATH] , on the other hand we have [MATH] which means all the sites are occupied by Kinesin heads. We therefore find that sixty percent of the sites are occupied by double-head attached Kinesins and forty percent by single-head attached Kinesins. In the ... |
We next turn into the issue of directed motion of Kinesins. First, we have computed the dependence of the averaged velocity of the CM versus [MATH] . Figure (11) sketches this behaviour. |
To obtain the above graph, we plotted [MATH] versus time and fitted a linear curve to it. The slope of the fitted line corresponds to the averaged velocity. As expected, [MATH] is a decreasing function of [MATH] . For small densities it decreases smoothly. However, when the density goes beyond |
[MATH] [MATH] shows a rapid decrease. In figure (12) we have exhibited the mean directed passage length (processive run length), the mean processive time and the velocity of this directed motion on the MT as a function of |
[MATH] The average processive run length [MATH] defined as the average distance a Kinesin can move proccesively on the MT before detachment exhibits a rapidly decreasing behaviour up to |
[MATH] . After this value it shows a weak dependence on [MATH] . We speculate that at this density large traffic jams are formed which do not allow Kinesins to move directedly. The analogous behaviour is observed for the mean temporal processivity [MATH] . Dividing [MATH] |
by [MATH] gives us the average velocity [MATH] of the processive motion. The average velocity has a more smooth decreasing behaviour. Having obtained the velocity of the directed motion, we are able to find the dependence of the current of Kinesins along the microtubule. The current [MATH] is defined by the fluid mecha... |
versus [MATH] [MATH] exhibits a maximum at [MATH] . The interesting point is the existence of an asymmetry in the [MATH] diagram. Normally in lattice driven gas models such as asymmetric simple exclusion process (ASEP) [MATH] appears as a symmetric function of [MATH] |
asep . The current [MATH] is related to the rate of cargo transport by Kinesin motor proteins. In order to find a deeper understanding, we have evaluated the steady state percentage of the bound and unbound Kinesins as a function of |
[MATH] . Bound Kinesins are divided into two groups: one-head attached and two-head attached to the MT. Figure (14) exhibits this behaviour. Percentage of unbound Kinesins increases with density. In the same manner, percentage of one-head attached Kinesins increases with [MATH] but it seems to become saturated at large... |
IV Summary and Concluding Remarks We have simulated the collective motion of Kinesins on a microtubular track by developing a two dimensional Langevin-type model. The model is capable of reproducing the hand-over-hand mechanism of the directed motion along the microtubule. Various quantities such as diffusion constant,... |
# Source: arxiv 1007.1026 # Title: Probabilistic initial value problem for cellular automaton rule 172 # Sections: all # Downloaded: 2026-03-02T08:58:20.927426+00:00 |
Probabilistic initial value problem for cellular automaton rule 172 (5 Apr 2010) Abstract We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the d... |
keywords: cellular automata, initial value problem, preimage trees Introduction While working on a certain problem in complexity engineering , that is, trying to construct a cellular automaton rule performing some useful computational task, the author encountered the following question. Let [MATH] be defined as |
[EQUATION] This function may be called selective copier , since it returns (copies) one of its inputs [MATH] or [MATH] depending on the state of the first input variable [MATH] . Suppose now that [MATH] be a bi-infinite sequence of binary symbols, i.e., [MATH] [MATH] . We will transform this string using the selective ... |
[MATH] otherwise, so that each [MATH] is simultaneously replaced by [MATH] . Consider now the question: Assuming that the initial sequence is randomly generated, what is the proportion of 1’s in the sequence after [MATH] iterations of the aforementioned procedure? |
Function defined by eq. ( ) is a local function of cellular automaton rule 172, using Wolfram numbering, and the aforementioned question is an example of a broader class of problems, which could be called probabilistic initial value problems for cellular automata: given initial distribution of infinite configurations ,... |
Basic definitions Let [MATH] be called a symbol set , and let [MATH] be the set of all bisequences over [MATH] , where by a bisequence we mean a function on [MATH] to [MATH] . Set [MATH] will be called the configuration space . Throughout the remainder of this text we shall assume that [MATH] , and the configuration sp... |
[MATH] will be simply denoted by [MATH] A block of length [MATH] is an ordered set [MATH] , where [MATH] [MATH] Let [MATH] and let |
[MATH] denote the set of all blocks of length [MATH] over [MATH] and [MATH] be the set of all finite blocks over [MATH] For [MATH] , a mapping [MATH] will be called a cellular automaton rule of radius [MATH] . Alternatively, the function [MATH] can be considered as a mapping of [MATH] into [MATH] |
Corresponding to [MATH] (also called a local mapping ) we define a global mapping [MATH] such that [MATH] for any [MATH] block evolution operator corresponding to [MATH] is a mapping |
[MATH] defined as follows. Let [MATH] be the radius of [MATH] , and let [MATH] where [MATH] . Then [EQUATION] Note that if [MATH] then [MATH] |
We will consider the case of [MATH] and [MATH] rules, i.e., elementary cellular automata . In this case, when [MATH] then [MATH] . The set |
[MATH] [MATH] will be called the set of basic blocks The number of [MATH] -step preimages of the block [MATH] under the rule [MATH] |
is defined as the number of elements of the set [MATH] Given an elementary rule [MATH] , we will be especially interested in the number of [MATH] -step preimages of basic blocks under the rule [MATH] |
Probabilistic initial value problem The appropriate mathematical description of an initial distribution of configurations is a probability measure [MATH] on [MATH] Such a measure can be formally constructed as follows. If [MATH] is a block of length [MATH] , i.e., |
[MATH] , then for [MATH] we define a cylinder set. The cylinder set is a set of all possible configurations with fixed values at a finite number of sites. Intuitively, measure of the cylinder set given by the block [MATH] , denoted by |
[MATH] , is simply a probability of occurrence of the block [MATH] in a place starting at [MATH] . If the measure [MATH] is shift-invariant, than |
[MATH] is independent of [MATH] , and we will therefore drop the index [MATH] and simply write [MATH] The Kolmogorov consistency theorem states that every probability measure [MATH] satisfying the consistency condition |
[EQUATION] extends to a shift invariant measure on [MATH] (Dynkin, 1969 .For [MATH] , the Bernoulli measure defined as [MATH] , where [MATH] is a number of ones in [MATH] and [MATH] is a number of zeros in [MATH] , is an example of such a shift-invariant (or spatially homogeneous) measure. It describes a set of random ... |
Since a cellular automaton rule with global function [MATH] maps a configuration in [MATH] to another configuration in [MATH] , we can define the action of [MATH] on measures on [MATH] For all measurable subsets [MATH] of [MATH] we define |
[MATH] , where [MATH] is an inverse image of [MATH] under [MATH] If the initial configuration was specified by [MATH] , what can be said about [MATH] (i.e., what is the probability measure after [MATH] |
iterations of [MATH] )? In particular, given a block [MATH] , what is the probability of the occurrence of this block in a configuration obtained from a random configuration after [MATH] iterations of a given rule? |
The general question of finding the iterrates of the Bernoulli measure under a given CA has been extensively studied in recent years by many authors, including, among others, |
Lind ( 1984 ); Ferrari et al. ( 2000 ); Maass and Martínez ( 2003 ); Host et al. ( 2003 ); Pivato and Yassawi ( 2002 2004 ); Maass et al. ( 2006 and Maass et al. ( 2006 In this paper, we will approach the problem from somewhat different angle, using very elementary methods and without resorting to advanced apparatus of... |
For a given block [MATH] , the set of [MATH] -step preimages is [MATH] . Then, by the definition of the action of [MATH] on the initial measure, we have |
[EQUATION] and consequently [EQUATION] Let us define the probability of occurrence of block [MATH] in a configuration obtained from the initial one by |
[MATH] iterations of the CA rule as [EQUATION] Using this notation, eq. ( ) becomes [EQUATION] If the initial measure is [MATH] , then all blocks of a given length are equally probable, and |
[MATH] , where [MATH] is the length of the block [MATH] . For elementary CA rule, the length of [MATH] -step preimage of [MATH] is [MATH] , therefore |
[EQUATION] This equation tells us that if the initial measure is symmetric ( [MATH] ), then all we need to know in order to compute [MATH] is the cardinality of [MATH] . One way to think about this is to draw a preimage tree for [MATH] . We start form [MATH] as a root of the tree, and determine all its preimages. Then ... |
Note that [MATH] corresponds to the number of vertices in the [MATH] -th level of the preimage tree. One thus only needs to know cardinalities of level sets in order to use eq. ( ), while the exact topology of connections between vertices of the preimage tree is unimportant. |
The key problem, therefore, is to enumerate level sets. In order to answer the question posed in the introduction, we need to compute [MATH] for rule 172, which, in turn, requires that we enumerate level sets of a preimage tree rooted at [MATH] . It turns out that for rule 172 the preimage tree rooted at 1 is rather co... |
[MATH] will exclusively denote the block evolution operator for rule 172. Block probabilities Since [MATH] , we have [MATH] . Due to consistency conditions (eq. ), |
[MATH] , and we obtain [EQUATION] This can be transformed even further by noticing that [MATH] , therefore [EQUATION] By using eq. ( ) and defining [MATH] we obtain |
[EQUATION] This means that in order to compute [MATH] , we need to know cardinalities of [MATH] -step preimages of 001, 101, and 111. |
Structure of preimage sets The structure of level sets of preimage trees rooted at [MATH] [MATH] , and [MATH] will be described in the following three propositions. |
Proposition 5.1 Block [MATH] belongs to [MATH] if and only if it has the structure [EQUATION] where [MATH] represents arbitrary symbol from the set [MATH] |
Let us first observe that [MATH] , which means that [MATH] can be represented as [MATH] . Similarly, therefore, [MATH] has the structure [MATH] , and by induction, for any [MATH] , the structure of [MATH] must be [MATH] [MATH] |
Proposition 5.2 Block [MATH] belongs to [MATH] if and only if it has the structure [EQUATION] where [MATH] for [MATH] and the string [MATH] does not contain any pair of adjacent zeros, that is. [MATH] for all [MATH] |
Two observations will be crucial for the proof. First of all, [MATH] , thus [MATH] has the structure [MATH] . Furthermore, we have |
[MATH] , meaning that if [MATH] appears in a configuration, and is not preceded by [MATH] , then after application of the rule 172, [MATH] will still appear, but shifted one position to the left. All this means that if [MATH] is to be an [MATH] -step preimage of [MATH] , it must end with [MATH] . After each application... |
Now, let us note that [MATH] , which means that preimage of [MATH] is either [MATH] or [MATH] . Therefore, we can say that if [MATH] is not present in the string |
[MATH] , it will not appear in its consecutive images under [MATH] . Thus, block [MATH] will, after each iteration of [MATH] , remain at the end, and will never be preceded by two zeros. Eventually, after [MATH] iterations, it will produce [MATH] , as shown in the example below. |
[EQUATION] What is left to show is that not having [MATH] in [MATH] is necessary. This is a consequence of the fact that [MATH] , which means that if [MATH] appears in a string, then it stays in the same position after the rule 172 is applied. Indeed, if we had a pair of adjacent zeros in |
[MATH] , it would stay in the same position when [MATH] is applied, and sooner or later block [MATH] , which is moving to the left, would come to the position immediately following this pair, and would be destroyed in the next iteration, thus never producing [MATH] . Such a process is illustrated below, where after thr... |
[EQUATION] Proposition 5.3 Block [MATH] belongs to [MATH] if and only if it has the structure [EQUATION] where [MATH] for [MATH] and the string [MATH] |
satisfies the following three conditions: (i) [MATH] for all [MATH] (ii) [MATH] and [MATH] (iii) if [MATH] , then [MATH] We will present only the main idea of the proof here, omitting some tedious details. It will be helpful to inspect spatiotemporal pattern rule 172 first, as shown in Figure . Careful inspection of th... |
(F1) A cluster of two or more zeros keeps its right boundary in the same place for ever. (F2) A cluster of two or more zeros extends its left boundary to the left one unit per time step as long as the left boundary is preceded by two or more ones. If the left boundary it is preceded by 01, it stays in the same place. |
(F3) Isolated zero moves to the left one step at a time as long as it has at least two ones on the left. If an isolated zero is preceded by 10, it disappears in the next time step. |
Let us first prove that (i)-(iii) are necessary. Condition (i) is needed because if we had [MATH] in the string [MATH] , its left boundary would grow to the left and after [MATH] iterations it would reach sites in which we expect to find the resulting string [MATH] |
Moreover, string [MATH] cannot have [MATH] at the end position, one site before the end, or two sites before the end. If it had, 0 preceded by two 1’s would move to the left and, after [MATH] |
iterations, it would reach sites where we want to find [MATH] . The only exception to this is the case when [MATH] . In this case, even if [MATH] is in the second position from the end, it will disappear in step [MATH] . This demonstrates that (ii) and (iii) are necessary. |
In order to prove sufficiency of (i)-(iii), let us suppose that the string [MATH] satisfies all these conditions yet [MATH] . This would imply that at least one of the symbols of [MATH] is equal to zero. However, according to what we stated in F1–F3, zero can appear in a later configuration only as a result of growth o... |
Enumeration of preimage strings Once we know the structure of preimage sets, we can enumerate them. For this, the following lemma will be useful. |
Lemma 6.1 The number of binary strings [MATH] such that [MATH] does not appear as two consecutive terms [MATH] is equal to [MATH] , where [MATH] is the [MATH] -th Fibonacci number. |
This result will be derived using classical transfer-matrix method. Let [MATH] be the number of binary strings [MATH] such that [MATH] does not appear as two consecutive terms [MATH] . We can think of such string as a walk of length [MATH] on a graph with vertices [MATH] and [MATH] which has adjacency matrix [MATH] giv... |
[MATH] [MATH] . One can prove that the generating function for [MATH] [EQUATION] can be expressed by [MATH] , where [EQUATION] and where [MATH] denotes the matrix obtained by removing the [MATH] row and [MATH] column of [MATH] . Proof of this statement can be found, for example, in Stanley ( 1986 Applying this to the p... |
[EQUATION] By decomposing the above generating function into simple fractions we get [EQUATION] where [MATH] is the golden ratio. Now, by using the fact that |
[EQUATION] and by using a similar expression for [MATH] , we obtain [EQUATION] where [MATH] is the [MATH] -th Fibonacci number, [MATH] This implies that [MATH] [MATH] |
Proposition 6.1 The cardinalities of preimage sets of [MATH] [MATH] [MATH] and [MATH] are given by [EQUATION] Proof of the first of these formulae is a straightforward consequence of Proposition 5.1 We have [MATH] arbitrary binary symbols in the string [MATH] , thus the number of such strings must be [MATH] |
The second formula can be immediately obtained using Lemma 6.1 and Proposition 5.1 . Since the first [MATH] symbols of [MATH] are arbitrary, and the remaining symbols form a sequence of [MATH] symbols without [MATH] , we obtain |
[EQUATION] In order to prove the third formula, we will use Proposition 5.3 . We need to compute the number of binary strings [MATH] satisfying conditions (i)-(iii) of Proposition 5.3 Le us first introduce a symbol [MATH] to denote the string of length [MATH] |
in which no pair [MATH] appears. Then we define: [MATH] is the set of all strings having the form [MATH] [MATH] is the set of all strings having the form [MATH] |
[MATH] is the set of all strings having the form [MATH] [MATH] is the set of all strings having the form [MATH] [MATH] is the set of all strings having the form [MATH] |
The set [MATH] of binary strings [MATH] satisfying conditions (i)-(iii) of Proposition 5.3 can be now written as [EQUATION] Since [MATH] are mutually disjoint, and [MATH] and [MATH] are disjoint too, the number elements in the set [MATH] is |
[EQUATION] which, using Lemma 6.1 , yields [EQUATION] Using basic properies of Fibonacci numbers, the above simplifies to [MATH] Now, since in the Proposition 5.3 the string [MATH] is preceded by [MATH] |
arbitrary symbols, we obtain [EQUATION] what was to be shown. Density of ones Using results of the previous section, eq. ( 11 ) can now be rewritten as |
[EQUATION] which simplifies to [EQUATION] or, more explicitly, to [EQUATION] Obviously, [MATH] , in agreement with the numerical value reported in Wolfram ( 1994 We can see that [MATH] converges toward |
[MATH] exponentially fast, with some damped oscillations superimposed over the exponential decay. This is illustrated in Figure , where, in order to emphasize the aforementioned oscillations, instead of [MATH] we plotted the ratio |
[EQUATION] as a function of [MATH] . One can show that [MATH] converges to the half of ratio divina (golden ratio), [MATH] , as illustrated in Figure . We can see from this figure that the convergence is very fast and that the agreement between numerical simulations and the theoretical formula is nearly perfect. |
Further results Results obtained in the previous two sections suffice to compute block probabilities for all blocks of length up to 3. Proposition 6.1 together with eq. ( ) yields formulas for |
[MATH] [MATH] , and [MATH] . Consistency conditions give [MATH] Furthermore [MATH] due to the fact that [MATH] Applying consistency conditions again we have [MATH] , hence |
[MATH] , and, similarly, [MATH] . This gives us probabilities of all blocks of length 2. Probabilities of blocks of length 3 can be obtained in a similar fashion: |
[EQUATION] The only missing probability, [MATH] is the same as [MATH] because [MATH] The following formulas summarize these results. |
[EQUATION] where [MATH] is the [MATH] -th Lucas number. We can also rewrite these formulas in terms of cardinalities of preimage sets using eq. ( ), as stated below. |
Theorem 8.1 Let [MATH] be the block evolution operator for CA rule 172. Then for any positive integer [MATH] we have [EQUATION] where [MATH] is the [MATH] -th Fibonacci number, [MATH] [MATH] , and [MATH] is the [MATH] -th Lucas number, [MATH] |
Concluding remarks The method for computing block probabilities in cellular automata described in this paper is certainly not applicable to arbitrary CA rule. It will work only if the structure of level sets of preimage trees is sufficiently regular so that the level sets can be enumerated by some known combinatorial t... |
One should add at this point that the convergence toward the steady state can be slower than exponential even in fairly “simple” cellular automata. Using similar method as in this paper, it has been found in Fukś and Haroutunian ( 2009 that in rule 14 the density of ones converges toward its limit value approximately a... |
As a final remark, let us add that the results presented here assume initial measure [MATH] This can be generalized to arbitrary [MATH] . In order to do this, one needs, instead of straightforward counting of preimages, to perform direct computation of their probabilities using methods based on Markov chain theory. Wor... |
# Source: arxiv 1007.3809 # Title: Segregation of receptor-ligand complexes in cell adhesion zones: Phase diagrams and role of thermal membrane roughness # Sections: all # Downloaded: 2026-03-03T05:15:26.470647+00:00 |
Segregation of receptor-ligand complexes in cell adhesion zones: Phase diagrams and role of thermal membrane roughness Abstract The adhesion zone of immune cells, the ‘immunological synapse’, exhibits characteristic domains of receptor-ligand complexes. The domain formation is likely caused by a length difference of th... |
Introduction Cell adhesion is mediated by the specific binding of a variety of membrane-anchored receptor and ligand molecules. In 1990, Springer suggested that the length difference of receptor-ligand complexes in the contact zone of immune cells may lead to segregation, i.e. to the formation of domains within the cel... |
. The ‘length’ of a receptor-complex here is the intermembrane distance, or local membrane separation at the site of the complex. A length difference between receptor-ligand complexes leads to an indirect, membrane-mediated repulsion of the complexes because the membranes have to bend to compensate the mismatch, which ... |
, the CD2-CD48 complex with the same length of 13 nm , and the LFA1-ICAM1 complex with a length of about 40 nm . In 1998 and 1999, the contact zone of T cells was indeed found to contain domains that either contain the short TCR-MHCp or the long LFA1-ICAM1 complexes |
. As expected from their length, the CD2-CD48 complexes are located within the TCR-MHCp domains . However, the question whether the domain formation is predominantly caused by the length mismatch of receptor-ligand complexes is complicated by the role of the actin cytoskeleton, which polarizes during T-cell adhesion an... |
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