A model of microtubule oscillations

Simulations of microtubule oscillations have been obtained by a kinetic model including nucleation of microtubules, elongation by addition of GTP-loaded tubulin dimers, disassembly into oligomers, and dissolution of oligomers followed by nucleotide exchange at the free dimers. Dynamic instability is described by the on and off rates for dimer association in the growth phase, the rate of rapid shortening, and the transition rates for catastrophe and rescue. The latter are assumed to be completely determined by the current state of the system (short cap hypothesis). Microtubule oscillations and normal polymerizations measured by time-resolved X-ray scattering were used to test the model. The model is able to produce oscillations without further assumptions. However, in order to obtain good fits to the experimental data one requires an additional mechanism which prevents rapid desynchronization of the microtubules. One of several possible mechanisms that will be discussed is the destabilization of microtubules by the products of disassembly.Abbreviations MT(s) microtubule(s) - G-MT/S-MT microtubule in the state of growth/shortening - GTP guanosine 5-triphosphate - GDP guanosine 5-diphosphate - TU · GDP/TU · GTP tubulin dimer with GDP/GTP bound to the exchangeable nucleotide binding site - MAP(s) microtubule-associated protein(s) - PC tubulin phosphocellulose-purified tubulin - PIPES piperazine-1,4-bis(2-ethane sulfonic acid) - DDT dithiothreitol - EGTA ethylene glycol-O,O-bis(2-amino ethyl ether)-N,N,N,N-tetraacetic acid     

Mechanochemical model of microtubule structure and self-assembly kinetics

Microtubule self-assembly is largely governed by the chemical kinetics and thermodynamics of tubulin-tubulin interactions. An important aspect of microtubule assembly is that hydrolysis of the beta-tubulin-associated GTP promotes protofilament curling. Protofilament curling presumably drives the transition from tip structures associated with growth (sheetlike projections and blunt ends) to those associated with shortening (rams' horns and frayed ends), and transitions between these structures have been proposed to be important for growth-shortening transitions. However, previous models for microtubule dynamic instability have not considered such structures or mechanics explicitly. Here we present a three-dimensional model that explicitly incorporates mechanical stress and strain within the microtubule lattice. First, we found that the model recapitulates three-dimensional tip structures and rates of assembly and disassembly for microtubules grown under standard conditions, and we propose that taxol may stabilize microtubule growth by reducing flexural rigidity. Second, in contrast to recent suggestions, it was determined that sheetlike tips are more likely to undergo catastrophe than blunt tips. Third, partial uncapping of the tubulin-GTP cap provides a possible mechanism for microtubule pause events. Finally, simulations of the binding and structural effects of XMAP215 produced the experimentally observed growth and shortening rates, and tip structure.     

Equilibrium states and oscillations for localized two-enzyme kinetics: A model for circadian rhythms

Two models of localized catalysis are presented. In Problem I we study cooperatively coupled enzymes bound to membranes separated from each other. In Problem II the interaction of a membrane bound with a soluble (homogeneously distributed on a finite interval) enzyme is considered, with non-linearity in the bulk solution. One of the models is developed to describe cellular circadian rhythms at the molecular level.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

Rapid microtubule self-assembly kinetics

Microtubule assembly is vital for many fundamental cellular processes. Current models for microtubule assembly kinetics assume that the subunit dissociation rate from a microtubule tip is independent of free subunit concentration. Total-Internal-Reflection-Fluorescence (TIRF) microscopy experiments and data from a laser tweezers assay that measures in vitro microtubule assembly with nanometer resolution, provides evidence that the subunit dissociation rate from a microtubule tip increases as the free subunit concentration increases. These data are consistent with a two-dimensional model for microtubule assembly, and are explained by a shift in microtubule tip structure from a relatively blunt shape at low free concentrations to relatively tapered at high free concentrations. We find that because both the association and the dissociation rates increase at higher free subunit concentrations, the kinetics of microtubule assembly are an order-of-magnitude higher than currently estimated in the literature.     

Tubulin oligomers and microtubule oscillations. Antagonistic role of microtubule stabilizers and destabilizers

Several types of non-equilibrium phenomena have been observed in microtubule polymerization, including dynamic instability, assembly overshoot and oscillations. They can be interpreted in terms of interactions between tubulin subunits (= alpha, beta heterodimers), microtubules, and a third state, oligomers, which represent intermediates between microtubule disassembly and the regeneration of assembly-competent subunits by GTP. Here we examine the role of oligomers by varying conditions that stabilize or destabilize microtubules and/or oligomers. By varying their ratio one can drive tubulin assembly either into steady-state microtubules or oligomers. These regimens of assembly conditions are separated by a region where microtubules oscillate. The oscillations can be simulated by computer modelling, based on a reaction scheme involving the three states of tubulin and nucleotide exchange on tubulin subunits, but not on microtubules or oligomers.     

A Hodgkin-Huxley model exhibiting bursting oscillations

We investigate bursting behaviour generated in an electrophysiological model of pituitary corticotrophs. The active and silent phases of this mode of bursting are generated by moving between two stable oscillatory solutions. The bursting is indirectly driven by slow modulation of the endoplasmic reticulum Ca²⁺ concentration. The model exhibits different modes of bursting, and we investigate mode transitions and similar modes of bursting in other Hodgkin-Huxley models. Bifurcation analysis and the use of null-surfaces facilitate a geometric interpretation of the model bursting modes and action potential generation, respectively.     

A model for the kinetics of crime

Michaelis-Menten kinetics was applied to processes of victimization. The dependence of the rate of victimization on population density was studied, and it was found that 56% of the variance in this complex social phenomenon may be explained with a one-variable model. The effect of police and the social significance of the rate constants are also discussed.     

Effects of glycerol on microtubule polymerization kinetics

Steady state and kinetic studies of polymerization of purified microtubule protein show little effect of glycerol on the steady state level of polymerization, as demonstrated by measurements of critical concentration. The rates of polymerization and depolymerization are slowed in the presence of glycerol. This data indicates that the stabilization of microtubules by high glycerol is largely a kinetic effect rather than a shift in equilibrium. However, the apparent critical concentration for microtubule polymerization from crude brain homogenate is substantially higher in the absence of glycerol, and glycerol appears to protect microtubule polymerization against the action of endogenous inhibitors.     

A composite model for establishing the microtubule arrays of the neuron

Neurons generate two distinct types of processes, termed axons and dendrites, both of which rely on a highly organized array of microtubules for their growth and maintenance. Axonal microtubules are uniformly oriented with their plus ends distal to the cell body, whereas dendritic microtubules are nonuniformly oriented. In neither case are the microtubules attached to the centrosome or any detectable structure that could establish their distinct patterns of polarity orientation. Studies from our laboratory over the past few years have led us to propose the following model for the establishment of the axonal and dendritic microtubule arrays. Microtubules destined for these processes are nucleated at the centrosome within the cell body of the neuron and rapidly released. The released microtubules are then transported into developing axons and dendrites to support their growth. Early in neuronal development, the microtubules are transported with their plus ends leading into immature processes that are the common progenitors of both axons and dendrites. This sets up a uniformly plus-end-distal pattern of polarity orientation, which is preserved in the developing axon. In the case of the dendrite, the plus-end-distal microtubules are joined by another population of microtubules that are transported into these processes with their minus-ends leading. Implicit in this model is that neurons have specialized machinery for regulating the release of microtubules from the centrosome and for transporting them with great specificity.     

A new kinetic model for biochemical oscillations: Graph-theoretical analysis

A graphical analysis demonstrates the ability of slow substrate activation and certain types of cooperativity between the two enzyme active sites to generate sustained oscillations. The analysis allows us to estimate kinetic parameter values for which oscillations exist. The scheme analyzed can explain the cyclical changes in functioning of various motor enzymes. Moreover, this scheme does not generate bistability for any parameter values. The graphical analysis presented is simple and visually clarifies the regulatory role of the details in the kinetic schemes.     

Microtubule oscillations. Role of nucleation and microtubule number concentration

Microtubules are capable of performing synchronized oscillations of assembly and disassembly which has been explained by reaction mechanisms involving tubulin subunits, oligomers, microtubules, and GTP. Here we address the question of how microtubule nucleation or their number concentration affects the oscillations. Assembly itself requires a critical protein concentration (Cc), but oscillations require in addition a critical microtubule number concentration (CMT). In spontaneous assembly this can be achieved with protein concentrations Cos well above the critical concentration Cc because this enhances the efficiency of nucleation. Seeding with microtubules can either generate oscillations or suppress them, depending on how the seeds alter the effective microtubule number concentration. The relative influence of microtubule number and total protein concentrations can be varied by the rate at which assembly conditions are induced (e.g. by a temperature rise): Fast T-jumps induce oscillations because of efficient nucleation, slow ones do not. Oscillations become damped for several reasons. One is the consumption of GTP, the second is a decrease in microtubule number, and the third is that the ratio of microtubules in the two phases (growth-competent and shrinkage-competent) approach a steady state value. This ratio can be perturbed, and the oscillations restarted, by a cold shock, addition of seeds, addition of GTP, or fragmentation. Each of these is equivalent to a change in the effective microtubule number concentration.     

A three-state model for connexin37 gating kinetics

The gating behavior of human connexin 37 (hCx37) is unaffected by the nature of the bathing monovalent (for Na, K, Rb). It is modified by [Mg] in the millimolar range. For fitting the kinetics, we propose a simple extension to three states of the canonical 2-state model of the hemichannel. The extra closed state allows for some immobilization of a hemichannel at high transjunctional voltages. The model is reasonably efficient at fitting data at various voltage protocols. Interpreting the fits of the data at different [Mg] is consistent with a binding site for Mg.     

A new model for binding of kinesin 13 to curved microtubule protofilaments

Kinesin motor proteins use adenosine triphosphate hydrolysis to do work on microtubules (MTs). Most kinesins walk along the MT, but class 13 kinesins instead uniquely recognize MT ends and depolymerize MT protofilaments. We have used electron microscopy (EM) to understand the molecular interactions by which kinesin 13 performs these tasks. Although a construct of only the motor domain of kinesin 13 binds to every heterodimer of a tubulin ring, a construct containing the neck and the motor domain occupies alternate binding sites. Likewise, EM maps of the dimeric full-length (FL) protein exhibit alternate site binding but reveal density for only one of two motor heads. These results indicate that the second head of dimeric kinesin 13 does not have access to adjacent binding sites on the curved protofilament and suggest that the neck alone is sufficient to obstruct access. Additionally, the FL construct promotes increased stacking of rings compared with other constructs. Together, these data suggest a model for kinesin 13 depolymerization in which increased efficiency is achieved by binding of one kinesin 13 molecule to adjacent protofilaments.