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Dynamics of Enzyme Digestion of a Single Elastic Fiber Under Tension: An Anisotropic Diffusion Model Abstract We study the enzymatic degradation of an elastic fiber under tension using an anisotropic random-walk model, coupled with binding-unbinding reactions that weaken the fiber. The fiber is represented by a chain o... |
pacs: 87.15.La, 87.15.Vv, 87.15.hg, 82.39.Fk The extracellular matrix (ECM), the biological structure that supports cells, is composed of elastic fibers such as elastin and collagen. The complex organization of these fibers undergoes a continuous maintenance that requires the catalytic action of enzymes, called proteas... |
The elasticity of a single fiber depends on how its molecular constituents are organized. During digestion, the molecules in the fiber as well as the cross-links can be cleaved by enzymes causing fiber stiffness to decreases. Furthermore, following cleavage, an enzyme can unbind, diffuse, bind at a different location a... |
In this letter, we study the decay of stiffness of a single fiber under tension during enzymatic digestion using an anisotropic random walk model coupled with binding-unbinding reactions. To our knowledge this is the first investigation of the mechanical properties of a single fiber that takes into account the simultan... |
Several different diffusion-reaction models have been used to describe processes at the level of ECM, cell membranes, macromolecules and DNA Abete_04 Dahirel_09 Schmit_09 Berry_02 Michael_05 Gennes_82 . In addition, the random walk is often used as a diffusion model that takes into account the morphological details of ... |
Our model consists of a one-dimensional chain of [MATH] linearly elastic springs in series representing an elastic fiber as in Fig. [MATH] . The fiber is surrounded by two layers of sites along which particles representing enzymes can diffuse. Periodic boundary conditions are applied in the [MATH] direction. Both ends ... |
In order to simulate enzyme activity on the fiber, we begin with a chain having identical initial spring constants [MATH] . The diffusion of enzyme is initiated by releasing a set of particles at random positions in the two layers. Each particle moves according to a set of probabilistic rules, controlling the diffusion... |
Local anisotropy is introduced through the probability [MATH] which depends on the local spring constant [MATH] [EQUATION] where [MATH] is the initial anisotropy and [MATH] is a characteristic length. Eq. ( ) expresses the fact that when the enzyme cleaves the fiber, the local [MATH] decreases by a small amount. Since ... |
The probability [MATH] is determined by the molecular properties of the specific enzyme and its substrate and is the same for each spring, for all times. A particle may remain bound for more than one time step, with probability [MATH] . We assume that cleavage occurs during unbinding so that [MATH] is reduced only when... |
Next, we study the evolution of [MATH] for different sets of parameters. We use a chain composed of [MATH] springs, different numbers of particles [MATH] and different values of the external force [MATH] within the interval [MATH] . The [MATH] and [MATH] are related to the experimentally controllable macroscopic parame... |
The results for [MATH] are plotted on a log-linear scale in Fig. . In all cases [MATH] shows two distinct exponentially decreasing regimes with time constants [MATH] and [MATH] separated by a crossover region around [MATH] . For fixed [MATH] [MATH] decreases monotonically as [MATH] increases. The [MATH] also decreases ... |
To characterize the changes in the microscopic properties of the fiber, we calculate the standard deviation [MATH] of all spring constants at a fixed time and average them over all runs in Fig. [MATH] . Initially, for [MATH] [MATH] increases quickly, which is not influenced by [MATH] . When [MATH] [MATH] starts to affe... |
To gain insight into the exponential decay of fiber stiffness, we carry out a simple mean field calculation. The local [MATH] at time [MATH] depends on the number of times the spring has been visited. We define [MATH] as the time corresponding to the [MATH] -th unbinding event along the entire chain. Thus, [MATH] at ti... |
[EQUATION] assuming [MATH] . Notice that [MATH] remains constant for [MATH] . At time [MATH] , an unbinding event occurs at spring [MATH] and the corresponding [MATH] is reduced to [MATH] . The new value of [MATH] is |
[EQUATION] where we also assume that [MATH] . Thus, from Eqs. ( ) and ( ), the change [MATH] in the total stiffness is written as |
[EQUATION] Next, we consider the average waiting time [MATH] between two unbinding events. During digestion, the number of particles [MATH] that remains bound on the fiber changes, but [MATH] is related to the number of free particles [MATH] since [MATH] . The rate of change of [MATH] is the difference between the aver... |
Since the unbinding probability is per unit time, [MATH] can now be expressed as [EQUATION] Finally, we can establish a link between the two processes involved in enzymatic digestion. From Eq. ( ), [MATH] is reduced by [MATH] during the interval [MATH] . We thus approximate the derivative of [MATH] by the discrete chan... |
[EQUATION] This equation can be solved assuming that [MATH] is approximately constant during one time step. The result is given by [MATH] , where |
[EQUATION] The above expression represents different aspects of the digestion process. The first term which involves [MATH] , is related to the geometry of the fiber, the average number of molecules in parallel. The second term is related to the specific enzyme activity at the microscopic level, the binding and unbindi... |
We next analyze the asymptotic limits of Eq. ( ). We summarize the results as follows [EQUATION] To compare the results of the numerical simulations with the analytical calculation in the asymptotic limits, we analyze the time course of stiffness by calculating the time constants in Fig. for the two different regimes o... |
In summary, we have presented a model for the enzymatic digestion of an elastic fiber under tension. We have shown that the total stiffness decreases exponentially with two different regimes separated by a crossover region. While the first exponential regime has been found experimentally Rajiv_07 , to our knowledge, th... |
# Source: arxiv 0912.1588 # Title: Self-organizing urban transportation systems # Sections: all # Downloaded: 2026-03-03T01:59:00.240353+00:00 |
Self-organizing urban transportation systems Abstract Urban transportation is a complex phenomenon. Since many agents are constantly interacting in parallel, it is difficult to predict the future state of a transportation system. Because of this, optimization techniques tend to give obsolete solutions, as the problem c... |
In this chapter, I will review recent, current, and future work on self-organizing transportation systems. Self-organizing traffic lights have proven to improve traffic flow considerably over traditional methods. In public transportation systems, simple rules are being explored to prevent the “equal headway instability... |
Introduction Traditional science, since the times of Galileo, Laplace, Newton, and Descartes, has assumed that the world is predictable (Kauffman, 2008 . The implications of this assumption can be clearly seen with Laplace’s demon: If an intellect had knowledge of the precise position and momentum of all atoms in the u... |
This does not imply that we should abandon all hope of predictability. In urbanism, as in other areas, it is certainly desirable to have a certain foresight before designing and building a system. However, we need to accept that our predictability will be limited. Knowing this, we can expect that the unexpected will co... |
A system designed as self-organizing focusses on building the components of the system in such a way that these will perform the function or reach the goal of the system by their dynamic interactions. Like this, if the goal of the system changes, the components will be able to adapt to the new requirements by modifying... |
In the next two sections, work on self-organizing traffic lights and on the equal headway instability phenomenon in public transportation systems is exposed, respectively. In Section future work is mentioned. A generalization of the use of self-organization in urban transportation systems closes the chapter. |
Self-organizing traffic lights Traffic lights in most cities are optimized for particular expected traffic flows. In some cities, there are different expected flows for different hours, e.g. morning rush hour, afternoon rush hour, low traffic, etc. The goal is to set green light periods and phases so that vehicles reac... |
An alternative lies in allowing traffic lights to self-organize to the current traffic situation, giving preference to streets with a higher demand (Ball, 2004 ; Gershenson, 2005 ; Cools et al 2007 ; Gershenson and Rosenblueth, 2009 . The main idea is the following: each intersection counts how many vehicles are behind... |
. A screenshot of a section of the simulation is shown in Figure The main results of our simulations of a simulated cyclic grid of ten by ten streets can be seen in Figure . For comparison, we also show results for a single intersection to indicate the capacities of an intersection in our model depending on vehicle den... |
The self-organizing method achieves free-flow, i.e. [MATH] for low densities. In other words, no vehicle stops. At medium densities, the maximum flux [MATH] is reached, i.e. intersections are used at their maximum capacity, i.e. a vehicle is always crossing an intersection. Gridlock is prevented at high densities, unle... |
The technology to implement the self-organizing method is already available. However, further details should be dealt with before the method can be implemented, e.g. how to include pedestrians. The method also offers potential advantages for handling vehicles with priority (public transport, emergency, police, etc.). W... |
Public transportation systems and the equal headway instability phenomenon Passengers in public transportation systems arriving randomly at stations are served best when the time intervals between vehicles—also known as the headway—is equal (Welding, 1957 , p. 133) . This is because equal headways imply regular interva... |
The equal headway instability phenomenon is present in most public transportation systems, including metros, trams, trains, bus rapid transit, buses, and elevators (Gershenson and Pineda, 2009 |
We have developed a simple model of a metro-like system and implemented it in a multi-agent simulation (available at ). We tested different constraints to promote equal headways. However, different parameters were best for different densities. As an alternative, we implemented adaptive strategies, where the parameters ... |
Even when these are encouraging results, the implementation of a technological solution is not enough, since transportation systems are used by people. The social aspect of the system cannot be neglected. In several systems, the main cause of the equal headway instability is passenger behavior. If appropriate measures ... |
For example, passengers should be discouraged from boarding crowded vehicles, since most probably they are leading a delayed platoon, with idling vehicles following behind. This could be encouraged with real-time information of the vehicle positions and/or expected time of arrival (one minute, ten minutes) and an indic... |
Other behaviors that should be promoted are those that allow a fast boarding and exit of passengers. Examples of these are letting people exit before entering and not standing near doors during a trip. The vehicles and stations can also be designed to facilitate these behaviors, e.g. having dedicated doors for entering... |
Future directions The approach used to coordinate traffic lights and to promote equal headways can be applied in other systems. For example: |
Coordination of public and private transport. The combination of the studies presented above can be useful to improve the performance of bus rapid transit systems (Levinson et al 2003 . These suffer from equal headway instability and are also affected by traffic lights. The implementation of self-organizing traffic lig... |
Highway traffic. Driving behavior can affect the capacity of highways. Different local rules and constraints can be explored to improve traffic flow. |
Crowd dynamics. Pedestrian behavior in crowded and panic situations can lead to undesirable situations (Helbing et al 2000 . Like with highway traffic, different changes in behavior can be explored to improve pedestrian flow and avoid accidents. |
Generalizing the use of self-organization The concept of self-orgnization can be used to design and build systems that are able to adapt to unforeseen situations in complex problem domains (Gershenson, 2007 . The main idea is to build the components of a system in such a way that they will find by themselves the soluti... |
Elements and systems can be described as agents with goals . We can assign a value to a variable [MATH] to represent the degree to which the goals of an agent have been met. The “ satisfaction ” of the agent is represented by [MATH] Agents in a complex system interact . These interactions can have positive, neutral, or... |
(Haken, 1981 . If the friction between local interactions is minimized, then the satisfaction of the system will be maximized (Gershenson, 2007 , p. 41) Mediators can be used to promote synergy and reduce friction. In this approach, the role of the designer lies in finding the appropriate mechanisms to steer agents in ... |
This methodology, detailed in Gershenson ( 2007 , was useful for developing the self-organizing traffic light controllers and the methods to promote equal headways. It can certainly be useful in other areas of urban transportation systems, as the ones mentioned in the previous section. For the case of the traffic light... |
Since the interactions in a complex system generate novel information, equation-based approaches are not sufficient for these problem domains. Multi-agent simulations are a complementary alternative, because interactions are generated as simulations are run. Statistical results of such simulations can give insights on ... |
Even when the focus here has been on urban transportation systems, the ideas presented could be applied in other areas of urbanism, e.g. adaptive urban planning and design. |
Acknowledgements I should like to thank Juval Portugali and his team for organizing the conference “Complexity Theories of Cities have come of Age”. Ideas on self-organization have been developed in collaboration with Francis Heylighen. Work on self-organizing traffic lights has been performed in collaboration with Seu... |
# Source: arxiv 1002.3907 # Title: Limits of sensing temporal concentration changes by single cells # Sections: all # Downloaded: 2026-03-03T05:15:08.490824+00:00 |
Limits of sensing temporal concentration changes by single cells Abstract Berg and Purcell [ Biophys. J. 20 , 193 (1977)] calculated how the accuracy of concentration sensing by single-celled organisms is limited by noise from the small number of counted molecules. Here we generalize their results to the sensing of con... |
Cells are able to sense concentration gradients with high accuracy. Large eukaryotic cells such as the amoeba Dictyostelium discoideum and the budding yeast Saccharomyces cerevisiae can sense very shallow spatial gradients by comparing concentrations across their lengths Arkowitz ( 1999 By contrast, small motile bacter... |
Sensing small numbers of molecules implies relative noise [MATH] , where [MATH] is the number of detected molecules. Berg and Purcell (BP) calculated how this noise affects the accuracy of concentration sensing Berg and Purcell ( 1977 . They considered three types of measurement devices: a single receptor, a perfectly ... |
A single receptor [Fig. (a)] binds particles at rate [MATH] and unbinds them at rate [MATH] . Following BP, we assume that diffusion is fast enough that the receptor never rebinds the same particle. An ideal observer has access to the binary time series [MATH] of receptor occupancy between [MATH] and [MATH] The lengths... |
[EQUATION] where [MATH] represents an ensemble average. Following a similar strategy, we can estimate the ramp rate by performing the linear regression of [MATH] to [MATH] |
[EQUATION] from which the concentration and the ramp rate are estimated using ( 31 ) as: [EQUATION] The uncertainties of these estimates can be calculated from the time correlations of receptor occupancy (see Appendix A.1.1 ), yielding: |
[EQUATION] where [MATH] is the total number of binding events in the time [MATH] . Note that the result for [MATH] is precisely that of BP Berg and Purcell ( 1977 ); Endres and Wingreen ( 2009 |
In Endres and Wingreen ( 2009 , it was shown that the accuracy of concentration sensing could be improved using maximum likelihood estimation. In this scheme, the parameters of the model are chosen to maximize the probability (“likelihood”) that the observed data was [MATH] can be characterized by the series of binding... |
[EQUATION] where [MATH] is the total bound time. The concentration and the ramp rate, [MATH] and [MATH] , are the model parameters. Given the times of the events, the likelihood is maximized with respect to [MATH] and [MATH] by solving |
[MATH] and [MATH] from which the maximum likelihood estimate [MATH] is obtained. In general these equations have no simple solution, but we can obtain the average behavior by exploiting the fact that binding and unbinding are fast with respect to concentration changes, i.e. that the receptor remains adiabatically in eq... |
[EQUATION] where [MATH] is the equilibrium occupancy at time [MATH] , given by ( 31 ) with [MATH] , where [MATH] and [MATH] are the true parameters that generated the data. Applying this approximation to [MATH] [MATH] , we confirm that [MATH] and [MATH] for [MATH] (see Appendix A.1.2 ). For finite times, the errors in ... |
[EQUATION] where [MATH] and [MATH] Again we can use the adiabatic approximation to compute the Hessian of the log-likelihood on the right-hand side of ( 34 ), to obtain: |
[EQUATION] These variances are half the ones obtained from LR ( ). The first result for constant concentrations is that of Endres and Wingreen ( 2009 As observed there, the LR estimate adds the uncertainties from both bound and unbound interval durations. In contrast, the maximimum likelihood estimate relies only on un... |
We now turn to ramp sensing by an entire cell, starting with the case of an idealized absorbing sphere [Fig. (b)]. An ideal observer witnesses a time series of absorption events, described by the instantaneous current [MATH] , where [MATH] is the Dirac delta function and [MATH] are the absorption times. The average cur... |
[EQUATION] The first result was obtained in Berg and Purcell ( 1977 . Maximum likelihood is difficult to implement in the context of the monitoring sphere because it requires a sum over all possible histories of particles exiting and returning to the sphere. Thus, whether the LR result can be improved upon remains an o... |
Maximum likelihood estimation is in general the optimal way to sense ramps, and provides a twofold improvement over simple linear regression in the case of the single receptor. Could MLE be implemented in biological systems? To address this question, we now introduce a simple, deterministic biochemical network (Fig. ) ... |
To extract the ramp rate from receptor activity requires a network that “takes the derivative” of its input signal. An example is the E. coli chemotaxis system, which relies on precise adaptation via integral feedback Barkai and Leibler ( 1997 ); Yi et al. ( 2000 . A minimal deterministic version of such a network is s... |
[EQUATION] where for simplicity [MATH] is the activity of a single receptor, and [MATH] is the concentration of signaling molecules it produces. |
[MATH] is a monotonically decreasing function regulating the production of [MATH] . The role of [MATH] is similar to that of the receptor methylation level in E. coli [MATH] precisely adapts the production rate of signaling molecules so that the steady-state value of [MATH] does not depend on the external ligand concen... |
[EQUATION] with [MATH] and [MATH] Thus [MATH] provides a readout of the absolute concentration, and [MATH] provides a readout of the ramp rate. The accuracy of these representations is limited by the ligand binding shot noise [MATH] . The effect of noise can be calculated by expanding the solution of ( 11 ) linearly ar... |
[EQUATION] [EQUATION] where [MATH] [MATH] can be imaginary). From ( 12 ) we deduce the uncertainties of [MATH] and [MATH] [EQUATION] |
For a fixed [MATH] , the optimal value of [MATH] is the smallest one with a non-oscillating response kernel [MATH] [MATH] . Systems with oscillating kernels are undesirable because they detect oscillations rather than ramps. For [MATH] our results are consistent with those of Eqs. ( ) and ( ), namely uncertainties inve... |
Despite its simplicity, our biochemical model may help analyze features of real biological systems. There are two separate aspects to the model: on the input side, different mechanisms of receptor signaling—continuous signaling (LR) versus burst signaling (MLE)—affect readout accuracy; on the output side, integral feed... |
Many receptors, including the well-studied chemotaxis receptors of E. coli , signal continuously rather than in bursts, and therefore do not employ MLE. In practice, how could cells implement MLE? A receptor could simply “store” a fixed amount of signaling molecules and release all of them upon ligand binding. Alternat... |
[MATH] the more peaked the distribution of [MATH] , the less noisy the readout. For equilibrium binding/unbinding, we find [MATH] (see Appendix ), with an irreversible binding cycle driven by energy dissipation required to achieve [MATH] Interestingly, there are examples of such irreversible cycles in ligand-gated ion ... |
As for the mechanism of ramp sensing, the integral feedback system underlying E. coli chemotaxis is similar to our simple model. However, the receptor methylation level, which plays the same role as [MATH] in our model, adjusts the binding/unbinding rates [MATH] so that [MATH] , rather than adjusting the production rat... |
We thank Pankaj Mehta and Aleksandra Walczak for helpful suggestions. T. M. was supported by the Human Frontier Science Program and N.S.W. by National Institutes of Health Grant No. R01 GM082938. |
Appendix A Uncertainties in ramp sensing A.1 Single receptor We consider a single receptor, which particles bind to and unbind from stochastically. The unbinding rate is denoted by [MATH] , and the binding rate by [MATH] , where [MATH] is the concentration of particles. We assume fast diffusion and so neglect rebinding... |
A.1.1 Linear regression We perform linear regression of the binary time series [MATH] of receptor occupancy [MATH] to [MATH] The time trace [MATH] can be characterized by the series of binding [MATH] and subsequent unbinding [MATH] times, [MATH] The resulting estimates for [MATH] and [MATH] are: |
[EQUATION] where [EQUATION] are the results of the linear regression. In the limit of long time series, these estimates give the correct answers: replacing [MATH] by its ensemble average value, [MATH] , where [MATH] [MATH] are the true values of the concentration and the ramp rate, one obtains: |
[EQUATION] which leads to [MATH] and [MATH] on average. The expected error can be obtained from the covariance matrix [MATH] To compute this covariance matrix we need the following quantity: |
[EQUATION] where the approximation [MATH] is valid provided that [MATH] . In the limit of large times we have: [EQUATION] from which we deduce (using Eqs. 14 15 and [MATH] ): |
[EQUATION] The above result for [MATH] was originally obtained by Berg and Purcell Berg and Purcell ( 1977 A.1.2 Maximum likelihood |
The probability (“likelihood”) of a sequence of binding and unbinding events at times [MATH] and [MATH] between [MATH] and [MATH] , is: |
[EQUATION] The log-likelihood is (up to a constant independent of [MATH] and [MATH] ): [EQUATION] Given the times of the binding and unbinding events, the optimal strategy for estimating the concentration and the ramp rate is to maximize this log-likelihood with respect to [MATH] and [MATH] i.e. to solve |
[EQUATION] from which we can obtain the maximum likelihood estimates [MATH] and [MATH] To solve Eq. 28 , we exploit the self-averaging property of [MATH] as [MATH] (adiabatic approximation). The sums in Eq. 26 and 27 become: |
[EQUATION] where [MATH] is the equilibrium probability of the receptor being bound at time [MATH] [EQUATION] and [MATH] is the true concentration. We take the derivatives of the log-likelihood (Eq. 27 , with replacements from Eqs. 29 and 30 ) with respect to [MATH] and [MATH] |
[EQUATION] It is straightforward to confirm that these expressions become zero for [MATH] and [MATH] Thus, in the limit of long time series, the maximum likelihood estimates coincide with the true values of the concentration and the ramp rate. The expected error of these estimates is given by the Cramér-Rao inequality,... |
[EQUATION] where [MATH] and [MATH] The second derivatives of the log-likelihood are, to leading order: [EQUATION] Exploiting the diagonal structure of the Hessian, we obtain: |
[EQUATION] where [MATH] is the total number of binding events in the time series, [MATH] . The above result for [MATH] was first obtained in Endres and Wingreen ( 2008 |
A.2 Absorbing sphere We now consider as a measurement device a perfectly absorbing sphere of radius [MATH] Solving Laplace’s equation with absorbing boundary conditions at the surface of the sphere yields the average flux of particles impinging on the sphere Berg and Purcell ( 1977 |
[EQUATION] where [MATH] is the diffusivity, [MATH] is the sphere radius, and [MATH] is the (time-dependent) concentration far away from the sphere. When the ramp arises from motion of the sphere up a spatial gradient (with velocity [MATH] ), one must be careful: in general the flux of particles is not uniform around th... |
[MATH] /s, [MATH] [MATH] m, and [MATH] [MATH] m/s, so that [MATH] [MATH] m/s. Therefore, from the perspective of a swimming bacterium, the concentration appears to be changing with time, essentially uniformly in space, which is the case we consider. |
A.2.1 Linear regression For a constant concentration, the simplest estimate for the concentration is [MATH] , where [MATH] is the total number of absorption events in time [MATH] In the presence of a ramp, an estimate of both concentration and ramp rate can be obtained by performing the linear regression of |
[EQUATION] to [MATH] , where the [MATH] are the absorption times, yielding: [EQUATION] [EQUATION] Since the particle absorptions are independent events, [MATH] is a Poisson variable, [MATH] (as above, [MATH] are the true values of the concentration and the ramp rate), and thus [MATH] Concerning the ramp rate, we have |
[EQUATION] and [EQUATION] where we have used the fact that over a short time [MATH] , the number of absorbed particles is a Poisson variable, which entails: |
[EQUATION] Therefore: [EQUATION] Finally, in summary: [EQUATION] where [MATH] is the total number of absorption events in time [MATH] |
A.2.2 Maximum likelihood The observations by an absorbing sphere in the interval [MATH] are summarized by the sequence of times when particles are absorbed, [MATH] [MATH] . The log-likelihood of this sequence reads: |
[EQUATION] where [MATH] is the expected average flux for concentration [MATH] and ramp rate [MATH] Setting the derivatives to zero, [MATH] [MATH] , yields to leading order: |
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