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EXPERIMENTAL SETUP AND METHODS Flow chamber The experiments have been carried out using a flow-chamber technique. A schematic of the set-up is shown in Fig. . A stainless-steel block and two coverslips are used to form a [MATH] mm chamber. Cells adhere on the bottom cover-slip. One duct of the chamber is connected to a... |
Cell culture PC12 cells are from the German Collection of Microorganisms and Cell Lines (DSMZ) 33 . They are plated on collagen coated slides and cultured in RPMI-1640 medium (Gibco) with 10% fetal bovine serum and 5% horse serum in presence of nerve growth factor (NGF) (Sigma-Aldrich Chemie, Munich, Germany) for 4–5 d... |
Experimental procedure Prior to an experiment, the slide with the adherent cells is transferred to the flow-chamber. Cells are allowed to stabilize for about 5 min by circulating the experiment medium (normal medium with addition of 25 mM HEPES buffer (Invitrogen, Darmstadt, Germany)). Experiments are performed by swit... |
Image analysis The volume and area of the neurites are analyzed from the recordings using a home-made edge detection program. Edge detection using a threshold for intensity is unreliable due to the “halo effect” present in phase-contrast images and also due to the dependence on the illumination intensity. To avoid such... |
Drug-induced cytoskeletal perturbation Experiments were performed in presence of cytoskeleton disrupting drugs in order to study the role of its individual components. A complication in these experiments arises due to the neurites becoming fragile or losing their cylindrical geometry on cytoskeletal perturbation. This ... |
EXPERIMENTAL RESULTS Volume dynamics We begin all experiments with a hypoosmotic shock imposed by switching the cell culture medium flowing through the chamber from normal medium with a total solute concentration [MATH] mM to a lower external value [MATH] (the continuous flow ensures constant external concentration at ... |
[MATH] Figure shows typical responses for three different values of [MATH] at 36 C. For weak shocks ( [MATH] ) the neurite volume increases from its initial volume [MATH] |
until it reaches a maximum steady value [MATH] . No recovery is observed for tens of minutes. For intermediate shocks, [MATH] , the volume increases at a roughly constant rate |
[MATH] initially until it reaches a maximum value [MATH] Subsequently, the volume recovers almost back to [MATH] with a typical regulatory volume decrease response (RVD). The volume recovery is roughly exponential with a characteristic time [MATH] . The recovery time [MATH] is strongly temperature-dependent (shown late... |
Once the volume stabilizes to the lower external osmolarity [MATH] (within about ten minutes), we perform a hyperosmotic shock by switching back the original medium with concentration [MATH] The neurite shrinks, reaches a minimum volume [MATH] and then comes back to its initial volume in an RVI response as shown in Fig... |
In the following, we separately address the volume regulation response and the peristaltic modulation. We study them as a function of osmotic shock strength, temperature, and in the presence of cytoskeleton-disrupting drugs. |
Nonlinear swelling response. Figure A shows the initial rate of change of volume divided by the initial area, [MATH] as a function of the initial osmotic pressure difference |
[MATH] for temperatures 33–36 C. Assuming zero hydrostatic pressure and neglecting changes in internal osmolarity gives the following expression for the initial swelling rate: |
[EQUATION] where [MATH] is the hydraulic permeability. Surprisingly, from the average [MATH] value at [MATH] we obtain an osmotic permeability [MATH] m/s (where [MATH] is the molar density of water), which is about two orders of magnitude lower than the literature values for lipidic membranes and most biological cells |
38 39 40 even after blockage of water channels 37 Moreover, as can be seen from Fig. A, the expected linear dependence is contradicted by the strong nonlinear response observed for both hypo- and hyperosmotic shocks. |
We now turn to the maximum (minimum) volume attained in a hypoosmotic (hyperosmotic) shock. The maximum (minimum) volume [MATH] is to a good approximation proportional to the initial volume [MATH] and does not depend significantly on the temperature (data not shown 41 ). Thus we look at the relative maximum volume [MAT... |
[EQUATION] where the dead volume [MATH] represents non-aqueous internal volume. Mammalian cells have on the average a cytosolic protein concentration of [MATH] |
. According to electron microscopy studies 30 the non-cytosolic volume of PC12 neurite is comprised mostly of microtubules and organelles and amounts to [MATH] The black region in Fig. corresponds to [MATH] for hypoosmotic shocks and to [MATH] for hyperosmotic shocks; penetrating this region would require work against ... |
Volume regulation under cytoskeleton disruption. The axonal cytoskeleton may be expected to contribute to the volume response in several ways. As discussed in the introduction, a mechanical as well as a signalling role is conceivable. In order to assess the role of individual components of the cytoskeleton, we treat ne... |
and the microtubule disrupting drug Nocodazole (NOC) 35 Since these are all diluted in dimethylsulfoxide (DMSO), a compound known to alter ion channels, we also perform control experiments in presence of DMSO. |
Fig. A shows typical responses. In presence of Nocodazole the initial swelling rate for strong shocks [MATH] increases markedly, but it barely changes for [MATH] |
(see Fig. B). With Blebbistatin we observe a weaker but still significant effect. For both drugs, the relationship between swelling rate and initial osmotic pressure difference approaches the naively expected linear dependence given by Eq. |
Nocodazole induced disruption of microtubules also has a strong effect on the maximum volume [MATH] attained after a strong shock of [MATH] as shown in Fig. C. Neither BLE-treatment nor DMSO alone have a significant effect on the maximum volume. For mild dilutions, [MATH] nocodazole has no effect, consistent with the f... |
Importantly, the cytoskeleton disrupting drugs do not affect the ability of the neurite to perform RVD. As can be seen in Fig. A, the volume fully relaxes back to its initial value. Further evidence can be found in the Supplementary Material. |
The effect of temperature: Arrhenius behaviour. We now address the influence of temperature in the dynamics of volume regulation. For simplicity, we describe the recovery phase by fitting single exponentials. As shown in Fig. lowering the temperature from 35 to 15 slows down the volume dynamics by an order of magnitude... |
[MATH] 30 kT, a typical order of magnitude for biological processes 42 The well-defined Arrhenius trend is consistent with the idea that ion channels are responsible for RVD. |
Beading When neurites are subjected to strong hypoosmotic shocks they undergo a shape transformation by developing a periodic array of swellings akin to beading of axons |
43 . We observe this in chick dorsal root ganglia (DRG) neurons as well as PC12 neurites Fig. . This peristaltic deformation, resembles that formed in nerves subjected to induced stretch injuries 31 32 44 and that observed after traumatic injuries to the brain |
43 The dynamics of bead formation and the mechanism has been investigated recently using the osmotic shock technique 20 and shown to be similar to the pearling instability observed in synthetic membrane tubes under tension 45 46 Here, we provide direct evidence correlating neurite tension and bead formation and describ... |
We begin by listing the main features of osmotic shock induced beading. For a given radius of the neurite, there is a critical hypoosmotic shock below which the shape remains cylindrical during the entire volume evolution, and above which a transient peristaltic modulation is observed. This is about |
[MATH] at 37 C and about [MATH] at 25 C for an initial neurite radius of 0.7 [MATH] m. Beading and recovery cycles (for mild shocks) can be repeated up to five times in the same neurite, after which neurites tend to detach from the substrate. Also transport of organelles can be observed during and after beading and rec... |
Axonal tension causes beading. The flow chamber technique has been used recently to quantitatively study the evolution of neurite tension in response to drag forces and to demonstrate active contractile responses in neurites 47 This is performed by imposing a constant, laminar flow perpendicular to the neurite generati... |
(see Fig. t=300s). The resulting strain gives a measure of the tension in the axon. Fig. shows snapshots of the neurite taken at various stages during an osmotic shock experiment. Under the influence of the flow the neurite attains an equilibrium catenary shape within about a minute. When the flow is switched from the ... |
Beading mechanism. The physical mechanism for bead formation is as follows 20 . After applying an osmotic shock the neurite volume increases as discussed above. This results in a corresponding expansion of the measured area [MATH] as shown in Fig. . This stretching of the membrane causes an increase in the tension of t... |
It can be shown 46 that at any instant [MATH] , for a volume [MATH] , a peristaltically modulated shape with [EQUATION] has a lower average surface area compared to a cylinder with the same instantaneous volume [MATH] , provided the wavelength [MATH] . For small amplitudes, the relative area gain can be obtained as |
[EQUATION] where S is the surface area and [MATH] . This can be seen in Fig. . The ratio [MATH] (note: the average volume and area are computed for unit length) increases when the peristaltic mode grows. This quantity is a constant for all cylinders irrespective of radius. An increase in [MATH] |
from this constant value indicates a decrease in area for the peristaltic shape as compared to a cylinder with identical volume. In other words, the neurite is able to reduce its interfacial energy by adopting a peristaltic shape instead of a cylindrical one. As mentioned earlier any deformation costs bulk elastic ener... |
The above mentioned expression for relative gain in area shows that reduction in area increases with wavelength of the perturbation, giving a maximum area gain for [MATH] |
[MATH] the length of the neurite. Clearly, the observed value of [MATH] is much shorter than and independent of the neurite length, but increases linearly with radius. A simple minded argument for this is as follows. Any perturbation with a wavelength [MATH] generates as pressure difference given by the Laplace law, th... |
[MATH] -axis, with a curvature radius [MATH] For large wavelengths, [MATH] , the latter can be ignored. The pressure difference in between the crest and the trough of a peristaltic mode can be written in the small amplitude limit ( [MATH] ) as |
[EQUATION] As indicated by the negative sign, this is an unstable flow: the pressure difference will drive water from the troughs into the crests, thereby increasing the perturbation amplitude [MATH] . The flow rate and hence the growth of a given mode is in general proportional to the driving force |
[EQUATION] Shorter wavelength modes are thus faster to grow than longer wavelength ones. However, for very short wavelengths the curvature along the neurite axis becomes important. The respective pressure difference is given by [MATH] In contrast to the previous case, here the cylindrical state is stable: water flows f... |
[EQUATION] Therefore for very small wavelengths the stabilizing flow driven by the curvature along [MATH] always dominates, and there is no instability anymore. We conclude that there is a fastest mode at an intermediate, non-zero wavelength. This can be shown to be [MATH] |
49 In the very dawn of linear stability analysis, this argument was applied to the case of water jets by Lord Rayleigh 50 51 THEORY |
We model volume regulation in neurites taking into account both mechanical and osmotic driving forces. The neurite is characterised by its volume, [MATH] and the internal amounts of ionic species, [MATH] For simplicity we consider only potassium and chloride, by large the most important osmolites inside the cell. Since... |
[EQUATION] one of the two concentrations can be eliminated. The two variables of our model are the adimensional quantities [EQUATION] |
The flow of water through the membrane is given by [EQUATION] where [MATH] is the neurite area, [MATH] the hydraulic permeability, |
[MATH] the hydrostatic pressure, and the osmotic pressure difference is given by [EQUATION] in terms of external [MATH] and internal osmolite concentrations [MATH] The typical relaxation time for the volume is |
[EQUATION] where [MATH] kPa the osmotic pressure of normal medium. Taking [MATH] m Pa -1 -1 for the permeability (from Fig. A) we obtain [MATH] s. |
We model ion movement with a passive K -Cl cotransport following Ref. neglecting the effect of ion pumps since these are not relevant for the short-term RVD response. Chloride flux is given by the difference in chemical potentials, |
[EQUATION] We model the RVD response following the standard assumption of a volume-dependent permeability The permeability [MATH] must be zero at the initial volume [MATH] and non-zero at a significant departure. We decompose |
[EQUATION] into a typical order of magnitude [MATH] and an adimensional function [MATH] For the chloride relaxation time [EQUATION] |
we expect a value of 10–100 s taking typical values for ion conductivities The volume-dependent permeability function [MATH] jumps from zero to a finite value at a critical threshold volume; otherwise its form is not known and different expressions have been used in the literature 17 In our experiments the critical vol... |
[EQUATION] where [MATH] is the step function. Finally, assuming external ion concentrations to jump instantaneously at time [MATH] |
from [MATH] to [MATH] the full system can be written as [EQUATION] with the parameters [EQUATION] Taking the following physiological values for the ion concentrations |
[EQUATION] the adimensional constants become [MATH] [MATH] , and [MATH] Zero pressure model. The standard assumption in modelling volume regulation is to set the hydrostatic pressure to zero |
[EQUATION] With this ansatz we can solve the equations for the volume response [MATH] As shown in Fig. the model reproduces the timescales for swelling and recovery as well as the maximum volume. Interestingly it reproduces the “undershoot” at [MATH] |
(see Fig. ). However, the dependence of the swelling rate on osmotic pressure is clearly off. The model gives a linear response –a feature intrinsic to any volume-dependent permeability– whereas experiments show a nonlinear one (Fig. A). |
Viscoelastic model. As shown by our experiments combining hypoosmotic shocks and drag forces (Fig. ), neurites develop tension while swelling. This suggests that the nonlinear response observed in experiments may come from a nonlinear mechanical response, a ubiquitous feature of cells with structured cytoskeleton 52 We... |
[MATH] is not zero, but depends on the swelling rate as expected for a viscoelastic element. To reproduce Fig. A, we assume that neurites swell at zero pressure below a critical rate but encounter internal friction for rates larger than a critical value [MATH] This can be written as |
[EQUATION] where [MATH] is the maximal rate of change of volume at which the neurite can swell without tension, corresponding to about 5 [MATH] in our experiments, and [MATH] is the friction scale, about 2 MPa. |
As shown in Fig. with this ansatz the model reproduces the essentially constant swelling rate observed in experiments. The nonlinear friction slows down both swelling and recovery. Interestingly, the peculiar triangular shape of of the curves at [MATH] is much closer to that of experiments (compare to Fig. ). |
DISCUSSION Microtubules mechanically slow down swelling. The initial response of neurites to hypo- as well as hyperosmotic shocks is a strongly nonlinear function of the external osmotic pressure. Remarkably, this response becomes much simpler after microtubule disruption: neurites behave as perfect-osmometers, and the... |
The mechanical response of the neurite, as described by Eq. 12 , seems to be similar to that of adhering fibroblasts 52 54 . At slow strain rates, forces are low; above a critical strain rate, friction increases and the force becomes much stronger. From our results, writing the critical rate as a strain we get |
[MATH] In single fibroblasts, this is indeed the order of magnitude of the critical strain rate where frictional forces increase 52 which in turn agrees with the timescale of active processes 11 This suggests that changes in cell shape –including swelling– take place at the rate allowed by spontaneous unbinding of cyto... |
RVD is osmotically driven. Our results indicate that mechanical tension does not provide the driving force for volume recovery. The pearling modulation vanishes well before the volume recovers, indicating zero membrane tension during RVD. Moreover, in presence of cytoskeleton-perturbing drugs the volume recovery time [... |
RVD follows an Arrhenius trend as a function of temperature. The timescales involved in the volume responses are temperature dependent. The relaxation time is about an order of magnitude more temperature dependent than the swelling/shrinking rate, showing an exponential reduction as the temperature is increased. Thus t... |
A dynamic picture of stretch injury. Similar shape transformations have been observed in nerves under the name of “beading” as a response to stretch injury 31 32 Interestingly, electron microscopy observations of the ultrastructure of stretch-beaded nerves show that microtubules are splayed out in the beads 44 consiste... |
Conclusions Neurites respond to sudden osmotic pressure changes with a fast volume regulation response. The initial phase is characterised by a nonlinear dependence of swelling rate on the initial osmotic pressure. Cytoskeletal perturbation, especially microtubule disruption, accelerates swelling and increases the maxi... |
Acknowledgements Experiments were performed at the Universität Bayreuth, Germany, with the generous financial support of Albrecht Ott. We thank Osvaldo Chara, Pablo Schwarzbaum, Karina Alleva and Wolfram Hartung for helpful discussions. |
# Source: arxiv 1006.4055 # Title: Computational modelling of the collective stochastic motion of Kinesin nano motors # Sections: all # Downloaded: 2026-03-03T05:15:22.166054+00:00 |
Computational modelling of the collective stochastic motion of Kinesin nano motors Abstract We have developed a two dimensional stochastic molecular dynamics model for the description of intra cellular collective motion of bio motors, in particular Kinesins, on a microtubular track. The model is capable or reproducing ... |
Introduction Various complex functions of Eucariotic cells namely mitosis, intracelluler transports and cell motility are based on the cooperative motion of molecular motor proteins such as Dyneins, Kinesins and Myosins howard schliwa1 schliwa2 ajdari These molecular motors move on cytoskeletal filaments like microtubl... |
schad1 nedelec1 nedelec2 schnitzer okada . A large number of in-vitro experiments have revealed that the motion of Kinesin is processive i.e., it performs stroke-type steps on its track before detachment 1p 2p . It is now well established that the Kinesins motion is derived by the free energy released in the chemical h... |
Due to stochasticity in the Kinesin motion, it would be reasonable to incorporate degrees of randomness in the modelling approach. In this spirit, a new approach namely asymmetric particle hopping models in continuous time which are based on master equation approach came into play klump2 klump5 kolomeisky1 frey Another... |
II Modelling of Kinesins movement The model we have considered is a generalisation of the model proposed in jamali for the motion of a single Kinesin. |
We shall now explain our stochastic ratchet model in some details. The force acting on each Kinesin head consists of four terms as follows: |
[EQUATION] [MATH] is the stochastic ratchet potential which models the interaction between Kinesin and microtubule. [MATH] is a bistable potential that models the elastic coupling between two heads of Kinesin, [MATH] is the repulsive potential between adjacent Kinesins and finally [MATH] represents both the stochastic ... |
II.1 Ratchet potential The ratchet potential [MATH] should involve an asymmetry along the MT. More precisely, we have taken the form of ratchet potential as follows |
jamali [EQUATION] With periodicity in [MATH] direction i.e., [MATH] where [MATH] is the microtubule period which is about [MATH] For [MATH] we have used the first eleven terms of a Fourier series of an asymmetric function [MATH] as follows: |
[EQUATION] The coefficients [MATH] and [MATH] are Fourier coefficients. For the asymmetric function [MATH] we chose the saw-tooth function: |
[EQUATION] [EQUATION] Where [MATH] denotes the integer part. The summation form of [MATH] has been used for computational convenience. Figure (1) shows [MATH] versus [MATH] |
Each peak of [MATH] is located at a distance [MATH] from the position of the next minimum. For more details see Ref. jamali . Concerning the part of potential [MATH] |
responsible for the perpendicular direction to the MT we have used the following potential : [EQUATION] where [MATH] . It is assumed that the above form of |
[MATH] can model the interaction between the molecules making up the Kinesin head and the MT. The symmetry of the function allows for detaching both in upward and downward directions perpendicular to the MT. This function is smoother than the Morse potential therefore we can choose larger time steps which can speed up ... |
Now we shall discuss the time dependent terms i.e., [MATH] and [MATH] of the Ratchet potential. The stochastic variable [MATH] specifies the biochemical affinity of each Kinesin head. It is a flashing potential which can switch between two discrete values [MATH] The value of [MATH] depends on the biochemical status of ... |
is set at [MATH] . The value of constant [MATH] determines the depth of the potential well. Its size must be such that the energy difference between the on-state (A=1) and the off-state (A=0) under the equilibrium condition is comparable with the energy released from the hydrolysis of the ATP i.e., |
[EQUATION] II.2 Viscous and stochastic thermal noise terms The force [MATH] , which for simplicity acts in two dimension, has the following structure: |
[EQUATION] [EQUATION] The first term corresponds to energy dissipation due to viscosity of the surrounding fluid. [MATH] and [MATH] are stationary random Gaussian white noises with zero mean. More precisely we have: |
[EQUATION] The drag coefficient [MATH] for a Kinesin in an aqueous environment is reported at [MATH] viscek . We have taken this value for [MATH] |
in this paper. II.3 The bistable potential The bistable potential is responsible for the elastic coupling of the two Kinesin heads. This potential is expressed by the following form jamali |
[EQUATION] In which [MATH] is the distance between the two heads, [MATH] is the amplitude of the potential and represents the coupling strength between heads, and [MATH] is the distance between the two potential minima. In the following figure, we have sketched the dependence of the bistable potential versus distance. |
II.4 The potential between different Kinesins heads In order to model the interaction between adjacent Kinesins, we introduce a short-ranged repulsive potential between heads of different Kinesins. More concisely, we have used the shifted-force repulsive part of the Lennard-Jones potential as follows: |
[EQUATION] In which [MATH] with [MATH] and the cut off length [MATH] . This repulsive potential prevents the heads of adjacent Kinesins from getting too close to each other. |
II.5 simulation details We have not yet discussed in details how quantities [MATH] an [MATH] vary with time. To this end, we introduce a quantity [MATH] for each Kinesin’s head which specifies whether at time [MATH] the head is attached to the MT or not. [MATH] is one if the head is attached to a binding site and zero ... |
[MATH] centred at a binding site of the MT then we consider this head as an attached one. Not we turn into [MATH] . Apparently the biochemical cycles of each head can be described by transition rates from one state to the other one. We assume that the average time that each head spends in the states K, K.ATP or K.ADP.P... |
[MATH] 14 16 . Since the rate of ATP consumption is considerably reduced when a head is detached 17 18 19 , in our model we have assumed that only an attached head received an ATP molecule. Moreover, we experimentally know the rate at which an attached head makes a transition from the state K |
[MATH] K.ATP strongly depends on the attaching state and position of its partner head 19 20 . In case the partner head is attached and rear, the above rate is lowered about two orders of magnitude. We have tuned the corresponding rates in our code such that the processivity and the average velocity of a single Kinesin ... |
[MATH] . The system has been simulated for [MATH] timesteps in each run. The first [MATH] timesteps are discarded for reaching to equilibrium. Kinesin global density [MATH] is defined as the number of Kinesins per binding site of the MT. Certainly [MATH] can exceed one. |
III simulation results Generically the Kinesin movement can be classified into 3 states: resting, forward proccesive movement and detachent/atachement to the MT. Once a Kinesin leaves the box from the vertical boundaries at [MATH] its corresponding image re enters from the opposite side. Those Kinesins which leave the ... |
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