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The largest possible extension of a D-polymer is reached when all MinD is bound in one polymer. This is the case at [EQUATION] For high total MinD concentrations, [MATH] can come close to [MATH] , i.e. the D-polymer would cover the whole cell from pole to pole. To avoid further assumptions on what happens if a polymer ... |
III.1.2 MinE We assume that the MinE polymer nucleates on the tip of the MinD polymer and grows on top of it towards the pole. The differential equation describing the projected length of the MinE polymer has the same simple structure as the one for the MinD polymer: |
[EQUATION] The same conversion factor [MATH] is used since we assume that MinE monomers bind to MinD monomers of the helix one-to-one. The MinE-polymer always starts growing from the non-polar tip of the MinD-polymer (Fig. ). The non-polar end of the MinDE-polymer falls off the membrane (with a slower speed than E-elon... |
[EQUATION] As for MinD, we assume the total number of MinE monomers to be constant: [EQUATION] III.2 Switching We consider a stochastic description of switching between the three possible discrete states of the state variables [MATH] where the probability of switching depends on the cytosolic concentrations of MinD/E. ... |
As a simplified model, we first consider the limit in which the cooperativities go to infinity. More specifically, we define the nucleation probability as [MATH] . In the limit [MATH] , the stochastic switching becomes deterministic with the nucleation event occurring as soon as [MATH] reaches the nucleation threshold ... |
Note that this deterministic limit of our model is fundamentally different from the deterministic model of Drew et al. Drew et al. ( 2005 in that Drew et al. consider the mean field behavior of a population of filaments that switch between growing and shrinking states at average rates whereas we have individual filamen... |
III.3 Model summary Eqs. ( )–( ) together with the above-mentioned switching dynamics of the discrete state variables [MATH] represent a hybrid dynamical system Champneys and di Bernardo ( 2008 . In the rest of this article, we analyse this system and present both analytical as well as numerical results that can be int... |
A single MinD-polymer ‘life span’ includes nucleation, growth, capping (nucleation of the MinE-polymer), and disassembly as described for the deterministic case in Tab. |
The two poles can undergo alternating events of this type, which then constitutes an oscillatory solution. Such an oscillatory solution progresses in the following manner (cf. Fig. and Fig. for notation). A MinD polymer capped (at time [MATH] ) by a MinE polymer disassembles at one pole while the nucleation site at the... |
For a specific subclass of the described oscillatory solution we are able to derive an analytical description for relevant probability distributions. This subclass we call regular oscillations and it is defined by [MATH] . A section of it is shown in Fig. . This definition essentially means that the growing phase ( [MA... |
Table gives an overview and numerical values for the parameters we used in this model. Most of them are taken in ranges reported in the experimental literature. Some are adjusted to values that lead to reasonable results of the model. |
IV Results A hybrid dynamical system must be solved piecewise and care must be taken at points of discontinuity which, in this case, occur each time a polymer switches state. Within the appropriate parameter ranges and for a regular oscillation (as defined above), we need only consider the progression through three of ... |
[EQUATION] where [MATH] is the time of the most recent capping of the left or right D-polymer, respectively. The solution of Eq. for a growing D-polymer is dependent on the dynamic cytosolic concentration. During regular oscillations, a growing D-polymer only appears while the other D-polymer is disassembling (i.e., di... |
[EQUATION] Similar solutions can be found for the dynamics of the E-polymers. However, for the analytical treatment in this article, we restrict ourselves to the limit of fast E-ring formation: We assume that an E-polymer attains its steady state length right after its nucleation (i.e. after capping of the D-polymer). ... |
For the rest of this article, we will drop the ‘D’ superscript and denote the length of the D-polymer by [MATH] IV.1 Solution of the deterministic model |
As a basis for further discussions we briefly present the solution to the simplest version of the model described in the preceding section and previously addressed by Cytrynbaum and Marshall Cytrynbaum and Marshall ( 2007 . We consider the case of deterministic switching (see Subsec. III.2 ) and fast E-ring formation. ... |
The calculation and the equations for the map are given in App. . In Fig. the map is plotted for varying total concentrations of MinD and MinE. |
The intersection of the map with [MATH] shows three fixed points which in terms of the hybrid system correspond to: 1. A stable cytosolic solution: There are no polymers ( [MATH] ) and all the Min proteins are in the cytosol as monomers. |
2. An unstable oscillation (the map intersects the identity line with a slope larger than one). 3. Stable (slope is equal to zero) oscillations with constant amplitude (given in Eq. A.3 ). |
Depending on the initial conditions ( [MATH] ), the system converges to one of the two stable states, i.e. each parameter set that allows for an oscillatory solution also includes a solution where no polymer exists (the cytosolic solution). |
As can be seen from Fig. , the region of initial conditions that lead to a stable oscillatory solution is biggest when [MATH] is large and [MATH] is close to [MATH] . If [MATH] is too small or [MATH] is too high, the map does not intersect with the identity line anymore and only the cytosolic solution remains. This cor... |
In the stochastic version of the model, low probability switching events can, under certain circumstances, lead to a transition between episodes of oscillations (with relatively constant amplitude) and almost purely cytosolic states. We will investigate this in the following sections and will refer to Fig. as the limit... |
IV.2 Probability distributions for experimentally measurable quantities More realistic than the deterministic-threshold limit for nucleation and capping of the MinD polymer is stochastic switching that models the random nature of the underlying chemical reactions. From now on, we will use the stochastic rules for switc... |
As long as the system undergoes regular oscillations, conditional probability distributions can be derived for the times at which nucleation and capping occur. Suppose the polymer on the right disappears at time [MATH] (Fig. ) and, at that moment, the polymer on the left is shrinking and has length [MATH] . Given this,... |
Using the two conditional probability distributions for nucleation and capping, analytical expressions for the steady state probability distribution of polymer lengths at capping and other relevant quantities can be derived. Similar to the description of the deterministic system in terms of a map (Subsec. IV.1 ), we de... |
Only in the case of regular oscillations can the two stochastic processes, nucleation and capping, be separated and the probabilistic map (Eq. D.6 ) be derived. Crucial for obtaining regular oscillations are high cooperativities in both capping and nucleation. |
In the following, we present probability distributions for the amplitude and period of the oscillations obtained from numerical simulations . We compare these results to analytical results that we obtain by iterating the probabilistic map. |
IV.2.1 Distribution of oscillation amplitude and dependence on cooperativities Fig. shows the results of long simulation runs of the full MinDE polymer model with stochastic switching. Plotted is the distribution of MinD-polymer lengths at capping (the amplitude of the oscillation) and the time between consecutive capp... |
In experimental studies (both qualitative and quantitative Shih et al. ( 2002 ), large deviations in the oscillation amplitude and period are seldomly reported. For our model to reproduce this narrow distribution, high cooperativities in both nucleation and capping are crucial. Fig. shows that our model is especially s... |
Fig. shows the same simulation results as of Fig. 4(a) together with the analytical result from the numerical iteration of the probabilistic map (Eqs. D.6 and D.7 ). |
The plots confirm the applicability of our analytical solution as discussed above. For regular oscillations, the analytical result agrees well with the results from simulations. When cooperativities decrease, the increased probability at small polymer lengths is not accounted for by the analytical result. These are the... |
IV.2.2 Distribution of oscillation periods and dependence on total concentrations Another quantity that can be easily obtained from experiment is the period of the oscillations. In Fig. , we plot the probability distributions of the period [MATH] for different total concentrations of MinD and MinE as obtained from simu... |
At higher concentrations of MinD ( [MATH] ) and intermediate concentrations of MinE ( [MATH] ), the distributions are sharply peaked around a single value of [MATH] (see Fig. 6(a) ). This parameter regime leads to regular oscillations. The amplitude of the oscillations decreases monotonically with [MATH] , reflected in... |
A similar distribution is obtained if the total MinE concentration is chosen too high (shown in Fig. 6(b) [MATH] ) with capping of the D-polymers occurring quite early. For low concentrations of MinE, the instantaneous capping rate is reduced and capping of the D-polymer can occur late (cf. also Fig. 11(b) ). In extrem... |
Using the steady state probability distribution for [MATH] , an analytical integral expression for the probability distribution of the period [MATH] can be derived for the case of regular oscillations (see App. ). Fig. shows a comparison of this analytical result with simulation data. Since the analytical description o... |
To illustrate the range of validity of our analytical calculation, we compare the means of the numerically and analytically obtained probability distributions in Fig. |
The deviations in mean period in Fig. reiterate that the analytic expression is valid provided MinD concentration is high and MinE concentration is intermediate, ensuring regular oscillations. For small [MATH] , the analytical result is always an overestimate of the numerical result, since the analytical treatment does... |
Our results as shown in Fig. agree qualitatively with experiments. Overexpression of MinD has been shown to increase the oscillation period Raskin and de Boer ( 1999a and overexpression of MinE is known to disrupt the normal division placement de Boer et al. ( 1989 , probably due to destroying the oscillatory pattern (... |
IV.2.3 Beat-skipping Stochastic switching (nucleation and capping) can also lead to occasional skipping of beats, another property that is easily quantified experimentally. Regular oscillations are characterized by the asynchronous and alternating growth and disassembly of a polymer at each of the two poles of the cell... |
As a skipped beat is one way of deviating from regular oscillations, the results in Fig. recapitulate the observations described earlier in the context of Figs. , and . The high fraction of irregular capping events for low total MinD concentrations as well as for high total MinE concentration correspond to the second p... |
IV.3 Bistability and stochastic transitions As derived in Subsec. IV.1 , the deterministic system is bistable for a large range of parameters. Under certain conditions, stochastic nucleation and capping of polymers can lead to transitions between the two stable states: oscillatory episodes lasting for tens of periods a... |
During regular oscillations, a D-polymer is most likely capped at a length close to the peak of the respective conditional probability distribution (Fig. 11(b) in App. ). Due to the stochasticity of the capping process, there is a small probability of the D-polymer being capped before it reaches normal extension . For ... |
In order to get a better understanding of the stochastic transitions we consider the special case of deterministic nucleation of D-polymers and stochastic capping . For high values of [MATH] , the map in Fig. can be used as a rough guide for the stochastic transitions. In App. , we adjust this map to the specific case ... |
In Fig. 10 we show three maps obtained as in App. : One representing our standard parameter set [MATH] [MATH] (cf. Fig. 7(b) ) and the two parameter sets used in Fig. 10(a) and 10(b) (with [MATH] ). One can think of the stochastic system as producing a cloud of points in this map around the stable steady states. A tran... |
Discussion The mechanisms underlying the Min oscillations in E.coli are still subject to much debate among modellers since none of the models proposed so far capture all the properties and all the phenotypes displayed in wildtype and mutant cells (see Cytrynbaum and Marshall ( 2007 for a discussion). The most obvious q... |
We generalized a recently published model to allow for more realistic comparison with data by introducing and analyzing stochasticity. From a mathematical viewpoint, this model provides an example of a stochastic hybrid dynamic system, whose solution can be found analytically in the case of regular oscillations. Probab... |
High cooperativity in the nucleation of the Min-polymers (at least a power of 3 for the MinD and 6 for MinE) are important. Lower cooperativities allow too many events that are inconsistent with observations. Reported Hill coefficients for the nucleation of MinD on a lipid membrane are around 2 Mileykovskaya et al. ( 2... |
The model is relatively stable to changes in MinD concentration. MinD concentrations greater than [MATH] ensure regular oscillations. MinE concentration, on the other hand, is more finely constrained and must fall between 1.2 and [MATH] |
Our results offer an explanation for the variability in the presence of oscillations observed in experiments. A significant fraction of cells usually do not show oscillations in experiments or go from an oscillatory state into a cytosolic state during the course of observation (e.g. Raskin and de Boer ( 1999b ). Our mo... |
In a recent experimental study, Downing et al. Downing actively controlled the MinD dynamics in E.coli by changing cationic concentrations in the surrounding medium. They were able to stop, distort and restart the oscillations. Based on the bistability and stochastic transitions described here, we speculate that the io... |
Most other models Pavin et al. ( 2006 ); Meacci and Kruse ( 2005 ); Tostevin and Howard ( 2006 produce a similar dependence of the oscillation period on the total Min concentrations and a few Fange and Elf ( 2006 ); Kerr et al. ( 2006 report on the variability of the period or, as in our case, the full probability dist... |
The biggest difference between the reaction-diffusion-type models and the polymer model described here is that in the polymer model the spatial pattern of the Min oscillations is prescribed in the form of possible nucleation sites of MinD in the membrane. This assumption becomes most important when trying to model the ... |
Appendix A The deterministic version as a one-dimensional map In its simplest version (deterministic switching and fast E-ring formation, cf. beginning of Sec. IV ), our model allows for a fully analytical solution in terms of a discrete map. We choose to display the maximal length of the MinD polymer as a function of ... |
To derive the map, we consider the following case: At time [MATH] , the DE-polymer on the left side ( [MATH] ) is decaying ( [MATH] ) and the D-polymer on the right side ( [MATH] ) is growing. Without loss of generality, we choose [MATH] such that the DE-polymer on the left just reached the length [MATH] , which causes... |
We now have to distinguish three different cases: 1. The newly nucleated polymer on the left side gets capped immediately if [MATH] . This is the case if |
[EQUATION] Under this condition, the cytosolic MinE concentration will never drop below [MATH] . Any nucleating MinD polymer will therefore be capped immediately, which evolves into the cytosolic state (i.e. no polymer: [MATH] ). |
2. For slightly larger [MATH] , nucleation on the left side will happen right after the first polymer (left) completely disassembled ( [MATH] [MATH] ), as long as [MATH] . Using [MATH] in Eq. leads to an expression for the length of the D-polymer at the next capping: |
[EQUATION] 3. If [MATH] is even bigger ( [MATH] ), such that [MATH] , the new polymer on the left will nucleate at [MATH] . Its capping will then happen at the stable amplitude: |
[EQUATION] This is the amplitude of the stable oscillation and all consecutive cappings will happen at this amplitude, too. The piecewise map derived above is displayed for various MinD and MinE concentrations in Fig. |
Appendix B Conditional probability distribution function for nucleation times (regular oscillation) In the stochastic version of our model, the probability distribution function of the times one side of the cell is free of polymer can be derived analytically if the system is oscillating regularly (see Subsec. III.3 ). ... |
The probability for nucleation between times [MATH] and [MATH] is the probability of no nucleation until [MATH] and then nucleating in [MATH] [MATH] [MATH] is the cumulative distribution function. Without loss of generality, we assume [MATH] (i.e., the polymer on the left side just decayed completely at time [MATH] ) a... |
[EQUATION] The cytosolic D-concentration follows [MATH] for [MATH] (on the other side). For [MATH] , there would only be cytosolic MinD ( [MATH] ). We do not consider this case, since it is outside the regular oscillation pattern. |
Putting [MATH] from above into Eq. B.1 , one obtains [EQUATION] Fig. 11 shows typical shapes of this probability distribution function (for a given [MATH] ) for high cooperativity in nucleation and different concentrations of MinD. From Eq. B.2 , it is obvious that the dependence of the probability distribution on [MAT... |
Appendix C Conditional probability distribution function for capping times (regular oscillation) Equivalently to the preceding appendix, one can derive an expression for the conditional probability distribution function for the time of growth of a polymer. In Subsec. III.2 we introduced the instantaneous probability fo... |
We only consider the approximative case of fast E-ring formation (see Sec. IV ). The MinDE-polymer on the right side is assumed to be decaying and has length [MATH] at time [MATH] (cf. Fig. ). Without loss of generality we assume that the D-polymer on the left side nucleates and starts growing at time [MATH] . We now h... |
1. First, the concentration of MinE monomers in the cytosol is constant ( [MATH] ). 2. For [MATH] [MATH] grows linearly: [MATH] 3. |
After [MATH] [MATH] is constant again: [MATH] Putting these into the equivalent of Eq. B.1 gives the conditional probability distribution function for the time, at which the growing D-polymer on the left side gets capped and switches states (see Eq. C.1 – we generalise to [MATH] and drop the side-dependence; [MATH] [MA... |
Here, [MATH] and [MATH] with [MATH] . The condition on [MATH] only enters through the dependencies in [MATH] and [MATH] Fig. 11 shows typical probability distribution functions for capping (for a given [MATH] ) for high capping cooperativity and different MinE concentrations. The dependence of the probability distribut... |
Appendix D A map for the probability distribution Combining the two conditional probability distributions computed in Apps. and , one can derive a map for the probability distribution of the amplitude during regular oscillations. Here, we derive this map in the following form: [MATH] , i.e. the probability distribution... |
We start with computing a relation between the two probability distributions [MATH] and [MATH] [EQUATION] The conditional probability distribution [MATH] can be obtained from Eq. B.2 by replacing [MATH] and multiplying with [MATH] (to ensure correct normalisation under the variable transformation). For the limits of th... |
With that, we obtain: [EQUATION] In a similar fashion, [MATH] can be expressed as a function of [MATH] [EQUATION] The conditional probability distribution in Eq. D.3 can be obtained from Eq. C.1 when replacing [MATH] with the solution of the deterministic relation that connects [MATH] and [MATH] (Eqs. and ): |
[EQUATION] Solving this equation for [MATH] gives a function [MATH] that involves a LambertW function and can therefore not be expressed in terms of elementary functions. For correct normalisation, the new conditional probability distribution also has to be multiplied by the derivative [MATH] |
The integration limits are the minimal and maximal values of [MATH] , such that the order of events in Fig. (i.e. regular oscillations) is still fulfilled for a given [MATH] . The upper limit turns out to be [MATH] , whereas the lower limit can be found as the solution to the polymer growth equation with [MATH] . One f... |
[EQUATION] Putting Eq. D.1 and Eq. D.3 together, one obtains a map from one probability distribution into the next one, half a period later: |
[EQUATION] The integrals can be computed numerically and by iterating Eq. D.6 , a steady state probability distribution can be obtained. A typical starting distribution for the iteration would be a delta distribution or a uniform distribution and for typical parameter values, a steady state distribution is reached afte... |
This approach is valid, as long as the vast majority of events fall into the scheme depicted in Fig. , i.e. as long as [MATH] and [MATH] (regular oscillations – see Subsec. III.3 ). |
A more relevant quantity than [MATH] , the length of a polymer at the time of nucleation on the other side, is the capping length [MATH] , i.e. the amplitude of the oscillation. To obtain the probability distribution for this length, one needs to perform another integration: |
[EQUATION] The conditional probability distribution can be obtained from Eq. C.1 , when replacing [MATH] with the appropriate function of [MATH] and [MATH] that can be derived from a similar relation as Eq. D.4 . We obtain |
[EQUATION] The lower limit of the integration is the same as in the previous integration (Eq. D.5 ). For the upper limit one needs to find [MATH] such that [MATH] . This can be found by solving Eq. D.4 with [MATH] for [MATH] . In case this solution is larger than [MATH] [MATH] (Eq. ) is the upper limit. |
Appendix E Probability distribution function for the period in the oscillatory state Using the iterative approach described in the previous appendix, the steady state probability distribution function for the lengths of D-polymers at a given time point during regular oscillations can be found. Starting from this distri... |
From Fig. , half a period [MATH] (the time between two consecutive nucleations on opposing sides) is found to be [MATH] and therefore [MATH] [MATH] can be replaced by [MATH] and an additional probability distribution [MATH] can be introduced in order to obtain an equation involving only distributions that have already ... |
[EQUATION] [MATH] is given in Eq. B.2 , how to obtain [MATH] is described in the preceding appendix and a steady state distribution for [MATH] is also derived there (iterating Eq. D.6 ). The upper limits of the two integrals are the smaller of [MATH] for [MATH] and [MATH] is the equivalent of Eq. D.5 [MATH] |
For the probability distribution function of the period [MATH] , one more integration is needed: [EQUATION] Appendix F A limiting deterministic map for the case of high cooperativity |
For the case of deterministic nucleation and stochastic switching, we want to use the deterministic map as shown in Fig. as a rough guideline for the dynamics of the system. In order to find this underlying map, we need to find a replacement for the parameter [MATH] in the analytical expression of the map (Eqs. A.1 A.3... |
# Source: arxiv 0905.3690 # Title: Thermodynamic constraints on neural dimensions, firing rates, brain temperature and size # Sections: all # Downloaded: 2026-03-03T05:14:56.683346+00:00 |
”Journal of Computational Neuroscience”, in press. Thermodynamic constraints on neural dimensions, firing rates, brain temperature and size |
Jan Karbowski Sloan-Swartz Center for Theoretical Neurobiology, Division of Biology 216-76, California Institute of Technology, Pasadena, CA 91125, USA; |
Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, 02-109 Warsaw, Poland Email: jkarb@its.caltech.edu |
Abstract There have been suggestions that heat caused by cerebral metabolic activity may constrain mammalian brain evolution, architecture, and function. This article investigates physical limits on brain wiring and corresponding changes in brain temperature that are imposed by thermodynamics of heat balance determined... |
pump depends biphasically on frequency, which can lead to the biphasic dependence of brain temperature on frequency as well. Both the total power of sodium pumps and brain temperature diverge for very small fiber diameters, indicating that too thin fibers are not beneficial for thermal balance. For very small brains bl... |
Keywords : Metabolism; Heat balance; Brain size; Fiber diameter; Temperature; Wiring; Limits. Introduction Brain like any computational device (Landauer 1961; Bennett, 1982) or biological organ (Rolfe and Brown, 1997) dissipates energy (Siesjo, 1978; Ames, 2000; Clarke and Sokoloff, 1994). Recent studies on cerebral me... |
This article answers these questions by finding the power generated by sodium metabolic pumps and relating this power to the thermal and neuroanatomical properties of brain tissue. From this, we determine theoretical thermal bounds on fiber dimensions, and compare them with empirical data. These bounds enable us to est... |
Methods 2.1 Voltage and Na -K dynamics. Na and K are two major ions affecting neural membrane dynamics (Kandel et al, 1991; Koch, 1998) and its energetics (Astrup et al, 1981; Erecinska and Silver, 1989; Rolfe and Brown, 1997; Ames, 2000). The energy consuming Na /K pump affects membrane electrical properties, because ... |
[EQUATION] where [MATH] is the membrane potential, [MATH] is the membrane capacitance per unit area, [MATH] is neuron’s membrane surface area (primary axons and dendrites), [MATH] and [MATH] are Na and K |
reversal potentials, [MATH] and [MATH] are voltage-dependent Na and K conductances per unit membrane area, [MATH] is the leak conductance and [MATH] is the reversal potential corresponding to the leak current. The parameter [MATH] denotes neuron’s volume, [MATH] is the Faraday constant. The current |
[MATH] is the Na /K pump current given by [MATH] where [MATH] mM, [MATH] is the maximal pump current per membrane surface area, and |
[MATH] is the Hill constant. The symbol […] denotes intracellular ionic concentration, [MATH] is the synaptic current, and [MATH] [MATH] are voltage dependent proportionality parameters determined below. |
Voltage dependent conductances are represented by [MATH] and [MATH] where [MATH] and [MATH] are the maximal Na and K conductances. The gating variables [MATH] and [MATH] obey the standard kinetic equation [MATH] (for [MATH] ), and the fast variable [MATH] is set to its equilibrium value [MATH] . Voltage dependences of ... |
[MATH] [MATH] [MATH] , and for potassium channels [MATH] [MATH] , where [MATH] is expressed in mV. It is assumed that the neuron represented by Eq. (1) is embedded in the network of neurons firing with an average firing rate [MATH] The synaptic current of a single synapse is |
[MATH] , where [MATH] is the release probability, [MATH] is the maximal synaptic conductance (it is assumed that the rising phase of synaptic conductance is much faster that its decaying phase that is characterized by the time constant [MATH] ), and [MATH] is the reversal potential for synapses. Since the majority of s... |
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