text stringlengths 128 2.05k |
|---|
Numerical simulations We use the numerical method described above to simulate our model of cargo import. The goal is to verify the experimental results presented by Roth et al. |
Two experiments are presented in : in the first, the NLS cargo known to bind to the MTs is activated in a cell with intact cytoskeleton, while in the second experiment the cell is treated with the MT-depolymerizing agent nocodazole (NCZ) to create a MT-less environment. |
In silico we can easily turn on and off the advection term and compare cargo accumulation in the nucleus. Notice that this corresponds to permitting or not the association to the MTs, while depolymerization cancels the MTs thus changing as well the cell environment. The full consequences on the cell metabolism under de... |
The initial conditions are the concentrations of the single molecular species when cell is at rest (i.e. in the absence of external stimuli). Initial concentrations used in the simulation are reported in Table . In particular we assume that the activated cargo is initially concentrated in a peripheral zone of the cytop... |
In such a situation, the concentrations of the complexes are zero (i.e. no complex is formed before the stimulus activates the cargo). The experimentally observed RanGTP accumulation of RanGTP in the nucleus forms in the initial phase of simulation. |
The accumulation in time of cargo mass in the nucleus is shown in Figure The change in accumulation rate is evident, and confirms the behavior experimentally obtained in |
A snapshot showing the simulated cargo concentration in a spherical cell after 17 seconds is shown in Figure (right), where the accumulation jump across the NE is clearly visible. Figure shows the concentration after the same time lapse, this time obtained with a two dimensional computation to better appreciate the var... |
# Source: arxiv 0907.3297 # Title: Directed d-mer diffusion describing Kardar-Parisi-Zhang type of surface growth # Sections: all # Downloaded: 2026-03-02T08:58:14.810843+00:00 |
Directed [MATH] -mer diffusion describing Kardar-Parisi-Zhang type of surface growth Abstract We show that [MATH] -dimensional surface growth models can be mapped onto driven lattice gases of [MATH] -mers. The continuous surface growth corresponds to one dimensional drift of [MATH] -mers perpendicular to the [MATH] -di... |
pacs: 05.70.Ln, 05.70.Np, 82.20.Wt One of the simplest nonlinear stochastic differential equation set up by Kardar, Parisi and Zhang (KPZ) Kardar et al. ( 1986 describes the dynamics of growth processes in the thermodynamic limit. It specifies the evolution of the height function [MATH] in the |
[MATH] dimensional space [EQUATION] Here [MATH] and [MATH] are the amplitudes of the mean and local growth velocity, [MATH] is a smoothing surface tension coefficient and [MATH] |
roughens the surface by a zero-average Gaussian noise field exhibiting the variance [MATH] The notation [MATH] is used for the noise amplitude and [MATH] |
means the distribution average. The KPZ equation was inspired in part by the the stochastic Burgers equation Burgers ( 1974 , which belongs to the same universality class |
Forster et al. ( 1977 , and it became the subject of many theoretical studies Halpin-Healy and Zhang ( 1995 ); Barabási and Stanley ( 1995 ); Krug ( 1997 . Besides, it models other important physical phenomena such as directed polymers Kardar ( 1985 , randomly stirred fluid |
Forster et al. ( 1977 , dissipative transport van Beijeren et al. ( 1985 ); Janssen and Schmittmann ( 1986 and the magnetic flux lines in superconductors Hwa ( 1992 The equation is solvable in [MATH] |
Kardar ( 1987 but in higher dimensions approximations are available only. As the result of the competition of roughening and smoothing terms, models described by the KPZ equation exhibit a roughening phase transition between a weak-coupling regime ( [MATH] ), governed by the Edwards-Wilkinson (EW) fixed point at [MATH] |
Edwards and Wilkinson ( 1982 , and a strong coupling phase. The strong coupling fixed point is inaccessible by perturbative renormalization group (RG) method. Therefore, the KPZ phase space has been the subject of controversies and the value of the upper critical dimension has been debated for a long time. |
Using a directed polymer representation, the validity of a scaling hypothesis Doty and Kosterlitz ( 1992 and the two-loop RG calculation for [MATH] |
Frey and Täuber ( 1994 was confirmed and extended to all orders in [MATH] Lässig ( 1995 . These results provided an argument for an upper critical dimension [MATH] of the roughening transition, but the strong-coupling rough phase is not accessible by perturbation theory. Above [MATH] the scaling behavior in the rough p... |
Halpin-Healy ( 1990 ); Stephanov ( 1994 ); Lässig ( 1998 . In particular, assuming that height correlations exhibit no multiscaling and satisfy an operator product expansion, exact field-theoretic methods lead to rational number growth values in two and three dimensions Lässig ( 1998 Some theoretical approaches predict... |
Canet et al. ( 2009 . This is in contradiction with the numerical results Ala-Nissila et al. ( 1993 ); Ala-Nissila ( 1998 ); Marinari et al. ( 2000 2002 which predict the lack of an upper critical dimension. |
Mapping of surface growth onto reaction-diffusion system allow effective numerical simulations Hinrichsen and Ódor ( 1999 ); Ódor ( 2004 As a generalization of the [MATH] dimensional roof-top model |
Plischke et al. ( 1987 ); Meakin et al. ( 1986 and the [MATH] dimensional octahedron model Ódor et al. ( 2009a , we consider the deposition and removal processes of higher dimensional objects on [MATH] dimensional surfaces. We remind that in [MATH] dimensions a continuous surface line having no overhangs can be approxi... |
Ligget ( 1985 . This is a lattice gas Katz et al. ( 1984 ); Schmittman and Zia ( 1996 , where particles can hop on adjacent sites with asymmetric rates and hard-core exclusion. Its behavior is well known, and variations of ASEP (disorder, interactions … etc.) correspond to variations of [MATH] dimensional KPZ growth mo... |
We have extended the roof-top construction to [MATH] dimensions Ódor et al. ( 2009a by the introduction of octahedra having four slopes. The up edges in the [MATH] or [MATH] directions can be represented by ’ [MATH] ’, while the down ones by ’ [MATH] ’, and a surface element update is a generalized (Kawasaki) exchange ... |
Now we proceed with this kind of construction, considering the discrete slope variables in higher dimensions and generalize the simultaneous [MATH] (Kawasaki) exchange rule of them (Eq.(3) of Ódor et al. ( 2009a ) to [MATH] -dimensional updates |
[EQUATION] with probability [MATH] for attachment and probability [MATH] for detachment (see Fig. .c for the 3d case). It is well known Barabási and Stanley ( 1995 that the surface evolution of the deterministic KPZ growth are described also by the Burgers equation |
Burgers ( 1974 for growth velocities ,t) in the surface normal obeying [EQUATION] due to the transformation [MATH] In the forthcoming part we will prove that our microscopic model for [MATH] -mers in the continuum limit can be mapped onto the anisotropic version of Eq. ( ), similarly as shown in lower dimensions |
Plischke et al. ( 1987 ); Ódor et al. ( 2009a . The derivation is based on the formulation of the reduction of possible updates. Our surface model is represented by the discrete derivative elements: |
[MATH] [MATH] [MATH] … ( [MATH] ) at every lattice points. A generalized Kawasaki update ( ) is defined by a matrix [EQUATION] In [MATH] dimensions we define vectors of the slopes, the columns of ( ), analogously to one and two dimensions: |
[MATH] around the lattice point, which we select for deposition/removal update and set up a microscopic master equation [EQUATION] |
with the probability distribution [MATH] . Here the prime index denotes the state of [MATH] following the update ). The transition probability of [MATH] -s can be expressed as |
[EQUATION] with [MATH] parametrization, which formally looks like the Kawasaki exchange probability in [MATH] d, except the factor [MATH] which is necessary to avoid surface discontinuity creation in higher dimensions. This means that we update the slope configurations only if the values of all coordinates of the vecto... |
[EQUATION] where [MATH] and [MATH] are the unity and the cyclic permutation matrices respectively. The matrix [MATH] shifts each coordinate value to the next index value. Thus for [MATH] -s with mixed coordinate values, the vectors |
[MATH] or [MATH] possess zero elements. Therefore the determinant of [MATH] being the product of the diagonal elements, is zero in case of mixed coordinates and [MATH] in case of equal coordinates. |
For example a [MATH] update is prohibited when the slope vector is [MATH] , because [MATH] has one coordinate value of zero [EQUATION] |
By calling ’ [MATH] ’-s as particles and the ’ [MATH] ’-s as holes of the base lattice, their synchronous update can be considered to be a single step motion of an oriented [MATH] -mer in the bisectrix direction of the |
[MATH] [MATH] [MATH] , … coordinate axes. Thus [MATH] -mers follow one-dimensional kinetics, described by Kawasaki exchanges ( Directed [MATH] -mer diffusion describing Kardar-Parisi-Zhang type of surface growth ). To obtain a one-to-one mapping we update neighborhoods of the lattice points denoted by the green dots of... |
To derive Eq. ( ) first we have to average over the slope vectors [EQUATION] By calculating its time derivative using the master equation ) and the transition probabilities ( Directed [MATH] -mer diffusion describing Kardar-Parisi-Zhang type of surface growth |
[EQUATION] in which we filter out vectors of non-equal coordinates ( Directed [MATH] -mer diffusion describing Kardar-Parisi-Zhang type of surface growth (thus [MATH] is nonzero only if |
[MATH] we can obtain [EQUATION] analogously to one dimension Plischke et al. ( 1987 Here one can see the discrete first and second differentials of |
[MATH] corresponding to the operators of Eq. ( ) in the bisectrix direction of the axes. These differentials are one-dimensional, because the [MATH] -mer dynamics is one-dimensional. In principle one could derive a set of coupled Burgers equations for the particles in each direction in an isotropic way in accordance wi... |
Making a continuum limit in each direction and taking into account the relation of height and slope variables [MATH] ), we can arrive to the deterministic KPZ equation. The nonlinear term vanishes for [MATH] [MATH] ). The sign of the coefficient [MATH] of the nonlinear term can be interpreted as follows: For [MATH] pos... |
Since this derivation was applied just for the first one in the hierarchy of equations for correlation functions it does not prove the equivalence to the stochastic KPZ. Furthermore, the form of the noise term, which was not considered in our derivation, may also introduce differences. Although our surface model is spa... |
Here we investigate by numerical simulations this isotopic surface growth model via the one-dimensional directed migration of [MATH] -mers in the [MATH] -dimensional space. We have developed bit-coded algorithms for the updates ( ) and run it with [MATH] , such that randomness comes from the site selection only. Theref... |
different local slope configurations. However, due to the surface continuity we need only a few bits of a world (1 byte for [MATH] |
and two bytes for [MATH] ) for this purpose. This allows an efficient storage management in the computer memory and permits simulations of larger system sizes. The updates can be performed by logical operations, either on multiple samples at once, or on multiple (not overlapping) sites at once. Our bit-coded algorithm ... |
We performed dynamical simulations by starting from stripe ordered particle distributions. This corresponds to a flat surface with a small intrinsic width. The considered lattices gases had the maximum linear sizes [MATH] for [MATH] dimensions, respectively and periodic boundary conditions were applied. A single step o... |
Kawasaki [MATH] -mer update ( ). The time is incremented by [MATH] in units of Monte Carlo steps (MCs). Throughout the paper we will use this unit of time. |
We could exceed by magnitudes of order all previous numerical system sizes and simulation times. For example the largest five-dimensional simulations were done for [MATH] and [MATH] MCs Ala-Nissila et al. ( 1993 Our [MATH] simulations, where we have the good bulk/surface ratio: [MATH] , required 2GB memory size and a c... |
system could achieve [MATH] MCs Tang et al. ( 1992 Our largest [MATH] sized simulations reached [MATH] MCs. The longest runs for [MATH] passed the saturation at |
[MATH] MCs and the samples were followed up to [MATH] MCs. We run the these lattice gas simulations for [MATH] independent realizations for each dimension and size considered, and calculated |
[MATH] and the second moment [EQUATION] from the height differences at certain sampling times. The growth is expected to follow the Family-Vicsek scaling Family and Vicsek ( 1985 asymptotically, but due to the corrections it can be described by a power series |
[EQUATION] with the surface growth exponent [MATH] . For finite system, when the correlation length exceeds [MATH] , the growth crosses over to a saturation with the scaling law |
[EQUATION] characterized by the roughness exponent [MATH] In our case the intrinsic width of the initial state, which is represented by a zig-zag surface of width [MATH] (see Fig. ), results in a constant correction term. Thus we have [MATH] [MATH] |
and [MATH] [MATH] During our scaling analysis we dropped this contribution by subtracting [MATH] from the raw data and consider the next leading order correction as leading one. Furthermore we disregarded the initial time region [MATH] , when basically an uncorrelated random deposition occurs. The dynamical exponent [M... |
and in case of the Galilean invariance of an isotropic KPZ equation the [MATH] relation should also hold. Besides the extensive simulations we have performed careful correction to scaling analysis by calculating the local slopes of the exponents. The effective exponent of the surface growth can be estimated similarly a... |
[EQUATION] It was determined numerically for different discretizations: [MATH] , and we tried to fit it with the leading-order correction ansatz, which can easily be deduced from ( 16 ) (see Reis ( 2004 or Ódor ( 2004 |
[EQUATION] for [MATH] and before the saturation region. In other cases, such as ballistic deposition, which has a large unknown intrinsic width one can use another effective roughness exponent definition introduced in Reis ( 2004 |
We tested our method with the one-dimensional , exactly known case. Simulations were run on [MATH] sized system up to [MATH] MCs for [MATH] independent realizations. We determined the effective exponents |
[MATH] , which approaches [MATH] from below, in a perfect agreement with the leading-order correction form (see Figure ). The fitting with ( 19 ) on the local slopes data resulted in |
[MATH] and [MATH] Similarly to the time dependence we can analyze the size dependence following the saturation by determining the effective exponent of the roughness, which can be defined as the logarithmic derivative of 15 |
[EQUATION] The finite size scaling was done for systems of linear sizes in between [MATH] (discussed earlier) and [MATH] , which was [MATH] for 2d, |
[MATH] for 3d, [MATH] for 4d and [MATH] for 5d, respectively. To handle the boundary conditions effectively, system sizes of power of 2 were simulated. To get the asymptotic values we took into account all effective exponent points shown on Fig. and applied a leading order, linear fitting. The error margins of exponent... |
by measuring the relaxation time, i.e. the time needed to reach 90% of the saturation value. The asymptotic value is extrapolated by a linear fitting: [MATH] |
In two dimensions we estimated the growth exponent in the largest system sizes considered ( [MATH] (see Fig. ). Fitting in the [MATH] time window with the form ( 19 ) resulted in [MATH] |
and [MATH] , which is somewhat bigger than what was obtained by the largest known ( [MATH] ) sized simulations: [MATH] Tang et al. ( 1992 , and all other previous numerical estimates including ours Ghaisas ( 2006 ); Reis ( 2004 ); Ódor et al. ( 2009a This value conciliates with the [MATH] RG exponent of Lässig ( 1998 O... |
[MATH] and intermediate times, which are damped before saturation. In the one-dimensional ASEP model such oscillations are shown to be the consequence of density fluctuations being transported through a finite system by kinematic waves GMGB07 One can speculate that the slight final increase of [MATH] |
for the largest system sizes is just a fluctuation or oscillation effect, but we could not eliminate this overall tendency by increasing the statistics. Although the statistical fluctuations grow dramatically, as [MATH] |
the increase of the mean value is observable for each size [MATH] Our error-bar of [MATH] reflects this uncertainty. The width saturation values have been investigated for [MATH] We took into account the leading order correction to-scaling by the following Ansatz |
[EQUATION] but due to the larger error-bars we restricted it to a linear approximation: [MATH] The local slopes of the steady state values [MATH] and of |
[MATH] are shown on Fig. This provides [MATH] and [MATH] for the roughness and [MATH] , with the linear coefficient [MATH] for the dynamical exponent. This roughness exponent is in agreement with RG value Lässig ( 1998 , and somewhat bigger than the existing figures |
[MATH] Marinari et al. ( 2000 for [MATH] and [MATH] Tang et al. ( 1992 for [MATH] In three dimensions the local slope analysis for [MATH] |
results in [MATH] and [MATH] agreeing with the numerical results from the literature: [MATH] Tang et al. ( 1992 ); Ala-Nissila et al. ( 1993 |
[MATH] Marinari et al. ( 2000 . But our estimate is much higher than [MATH] Ghaisas ( 2006 (based on [MATH] sized simulations) and [MATH] predicted by RG Lässig ( 1998 |
For the saturation we obtained [MATH] and [MATH] matching [MATH] of Ghaisas ( 2006 and in marginal agreement with [MATH] of Marinari et al. ( 2000 and [MATH] of |
Ala-Nissila et al. ( 1993 . The direct [MATH] measurement exhibits a strong correction to scaling: [MATH] [MATH] ) and one cannot differentiate it from the [MATH] dimensional results within the error margins. |
In four spatial dimensions our best fit for the growth exponent is [MATH] and [MATH] . In the literature [MATH] Ala-Nissila ( 1998 and [MATH] |
Marinari et al. ( 2000 values are reported. For the width saturation values the linear fitting results in [MATH] with [MATH] This compares with the literature values [MATH] |
Ala-Nissila et al. ( 1993 and [MATH] Marinari et al. ( 2000 The [MATH] seems to converge to [MATH] (with [MATH] but the fluctuations are very strong and we could not reach saturation for sizes larger than [MATH] Going further by a factor of two in system sizes would require simulations with 8GB memory and very long CPU... |
In five dimensions the local slopes suggest [MATH] and [MATH] in agreement with [MATH] Ala-Nissila et al. ( 1993 reported for smaller sizes. One can find strong oscillations before the saturation regime. Again these are due to kinematic transport waves in finite system. Initially for [MATH] we saw a definite increase i... |
as [MATH] before the saturation, but this proved to be an artifact of the MT random number generator. When we used different, pseudo-random number generators: drand48 |
or random() of language C, the growth tendency for very late times was much weaker. We think that the site selection, the only source of randomness in case of [MATH] might not be completely uniform among the [MATH] possible places. To confirm this we repeated the [MATH] simulations using [MATH] with the MT generator an... |
and [MATH] with [MATH] In conclusion we have shown that the mapping of a KPZ surface growth model onto driven lattice gases (DLG) can be extended to higher dimensions. Although the growth of the surfaces exhibits the spatial symmetry of the underlying lattice, one can map it onto an anisotropic DLG of more complex obje... |
[MATH] -mers with hard-core exclusions. The topological constraint is the consequence of the required surface continuity by the mapping. In two dimensions we confirmed Ódor et al. ( 2009b that the probability distribution [MATH] matches the universal scaling function determined for another KPZ model Marinari et al. ( 2... |
[MATH] exponents in all dimensions. For [MATH] our results marginally overlap with the [MATH] value suggested some time ago by RG. The change towards a trivial behavior in higher dimensions in the DLG language would mean the disappearance of the topological constraints among the extended [MATH] -mer objects as they cou... |
We thank Zoltán Rácz for the useful comments. Support from the Hungarian research fund OTKA (Grant No. T77629), the bilateral German-Hungarian exchange program DAAD-MÖB (Grant Nos. D/07/00302, 37-3/2008) is acknowledged. G. Ódor thanks for the access to the Clustergrid and the NIIF supercomputer. |
# Source: arxiv 0908.0657 # Title: Role of ATP-hydrolysis in the dynamics of a single actin filament # Sections: all # Downloaded: 2026-03-03T05:15:03.462937+00:00 |
Role of ATP-hydrolysis in the dynamics of a single actin filament Abstract We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work,... |
Key words: Actin; ATP hydrolysis; Stochastic dynamics Introduction Actin monomers polymerize to form long helical filaments, by addition of monomers at the ends of the filament. The two ends are structurally different. The addition and removal of subunits at one end, the barbed end, are substantially faster than at the... |
. But actin is not an equilibrium polymer, it is an ATPase, and ATP is rapidly hydrolyzed after polymerization. Due to this constant energy consumption, the actin polymer exhibits many interesting non-equilibrium features; most notably it is able to maintain different critical concentrations at the two ends |
. This allows the existence of a special steady-state called treadmilling, characterized by a flux of subunits going through the filament, which has been observed both with actin as well as with microtubules filaments |
The precise molecular mechanism of hydrolysis in actin has been controversial for many years. For each of the two steps involved in the hydrolysis (the ATP cleavage and the Pi release), the possibility of the reaction occurring either at the interface between neighboring units carrying different nucleotides or at rando... |
A direct evidence for a cooperative mechanism was brought recently by the authors of Ref. , who observed GTP-tubulin remnants using a specific antibody. |
Several groups have emphasized the process of random cleavage followed by random Pi release . By studying the polymerization of actin in the presence of phosphate, the authors of Ref. argued that the crucial step of release of the phosphate is not a simple vectorial process but is probably cooperative. Since this relea... |
Although decades of work in the biochemistry of actin have provided a lot of details on the kinetics of self-assembly of actin in the absence and in the presence of actin binding proteins, it is difficult to capture the complexity of this process without a mathematical model to organize all this information. To this en... |
based on the work of Stukalin et al. 12 . In this model, the hydrolysis of subunits inside the filament is a vectorial process, the filament is in contact with a reservoir of monomers, and growth occurs only from one end. We have analyzed the phase diagram of that model with a special emphasis on the bounded growth pha... |
The work of Flyvbjerg et al. 14 has inspired a number of other theoretical models, based on a microscopic treatment of growth, decay, catastrophe and rescue of the filament: see in particular Ref. 15 and Refs. 16 17 18 , which analyze using analytical and numerical methods several aspects of the dynamic instability of ... |
In this paper, we present a model for a single actin filament which accounts for the insertion, removal, and ATP hydrolysis of subunits at both ends. It extends our previous work |
11 in several ways: first by including the dynamics of both ends and secondly by carrying out simulations for both mechanisms of hydrolysis – vectorial and random. In section |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.