Quantitative Biology

Length-regulation of microtubules (MTs) is essential for many cellular processes. Molecular motors like kinesin 8, which move along MTs and also act as depolymerases, are known as key players in MT dynamics. However, the regulatory mechanisms of length control remain elusive. Here, we investigate a stochastic model accounting for the interplay between polymerization kinetics and motor-induced depolymerization. We determine the dependence of MT length and variance on rate constants and motor concentration. Moreover, our analyses reveal how collective phenomena lead to a well-defined MT length.

This paper describes an investigation into part of the mechanical mechanisms underlying the formation of mitotic spindle, the cellular machinery responsible for chromosomal separation during cell division. In normal eukaryotic cells, spindles are composed of microtubule filaments that radiate outward from two centrosomes. In many transformed cells, however, centrosome number is misregulated resulting in cells with more than two centrosomes. Addressing the question of how these cells accommodate these additional structures by coalescing supernumerary centrosomes to form normal spindles will provide a powerful insight toward understanding the proliferation of cancer cells and developing new therapeutics. The process of centrosome coalescence is thought to involve motor proteins that function to slide microtubules relative to one another. Here we use in vitro motility assays combined with fluorescence microscopy to visualize, characterize and quantify microtubule-microtubule interactions. After segmenting the microtubules, their speed and direction of movement are the extracted features to cluster their interaction type. In order to evaluate the potential of our processing algorithm, we created a simulated dataset similar to the time-lapse series. Once our procedure has been optimized using the simulated data, we will apply it to the real data. Results of our analyses will provide a quantitative description of interaction among microtubules. This is a potentially important step toward more thorough understanding of cancer.

The disappearance of the guanosine triphosphate (GTP)-tubulin cap is widely believed to be the forerunner event for the growth-shrinkage transition (`catastrophe') in microtubule filaments in eukaryotic cells. We study a discrete version of a stochastic model of the GTP cap dynamics, originally proposed by Flyvbjerg, Holy and Leibler (Flyvbjerg, Holy and Leibler, Phys. Rev. Lett. 73, 2372, 1994). Our model includes both spontaneous and vectorial hydrolysis, as well as dissociation of a non-hydrolyzed dimer from the filament after incorporation. In the first part of the paper, we apply this model to a single protofilament of a microtubule. A catastrophe transition is defined for each protofilament, similar to the earlier one-dimensional models, the frequency of occurrence of which is then calculated under various conditions, but without explicit assumption of steady state conditions. Using a perturbative approach, we show that the leading asymptotic behavior of the protofilament catastrophe in the limit of large growth velocities is remarkably similar across different models. In the second part of the paper, we extend our analysis to the entire filament by making a conjecture that a minimum number of such transitions are required to occur for the onset of microtubule catastrophe. The frequency of microtubule catastrophe is then determined using numerical simulations, and compared with analytical/semi-analytical estimates made under steady state/quasi-steady state assumptions respectively for the protofilament dynamics. A few relevant experimental results are analyzed in detail, and compared with predictions from the model. Our results indicate that loss of GTP cap in 2-3 protofilaments is necessary to trigger catastrophe in a microtubule.

Microtubules are highly regulated dynamic elements of the cytoskeleton of eukaryotic cells. One of the regulation mechanisms observed in living cells is the severing by the proteins katanin and spastin. We introduce a model for the dynamics of microtubules in the presence of randomly occurring severing events. Under the biologically motivated assumption that the newly created plus end undergoes a catastrophe, we investigate the steady state length distribution. We show that the presence of severing does not affect the number of microtubules, regardless of the distribution of severing events. In the special case in which the microtubules cannot recover from the depolymerizing state (no rescue events) we derive an analytical expression for the length distribution. In the general case we transform the problem into a single ODE that is solved numerically.

The interaction between actin filaments and microtubules is crucial for many eukaryotic cellular processes, such as, among others, cell polarization, cell motility and cellular wound healing. The importance of this interaction has long been recognised, yet very little is understood about both the underlying mechanisms and the consequences for the spatial (re)organization of the cellular cytoskeleton. At the same time, understanding the causes and the consequences of the interaction between different biomolecular components are key questions for \emph{in vitro} research involving reconstituted biomolecular systems, especially in the light of current interest in creating minimal synthetic cells. In this light, recent \emph{in vitro} experiments have shown that the actin-microtubule interaction mediated by the cytolinker TipAct, which binds to actin lattice and microtubule tip, causes the directed transport of actin filaments. We develop an analytical theory of dynamically unstable microtubules, nucleated from the center of a spherical cell, in interaction with actin filaments. We show that, depending on the balance between the diffusion of unbound actin filaments and propensity to bind microtubules, actin is either concentrated in the center of the cell, where the density of microtubules is highest, or becomes localized to the cell cortex.

We formulate a dynamical model for microtubules interacting with a catastrophe-inducing boundary. In this model microtubules are either waiting to be nucleated, actively growing or shrinking, or stalled at the boundary. We first determine the steady-state occupation of these various states and the resultant length distribution. Next, we formulate the problem of the Mean First-Passage Time to reach the boundary in terms of an appropriate set of splitting probabilities and conditional Mean First-Passage Times, and solve explicitly for these quantities using a differential equation approach. As an application, we revisit a recently proposed search-and-capture model for the interaction between microtubules and target chromosomes [Gopalakrishnan & Govindan, Bull. Math. Biol. 73:2483--506 (2011)]. We show how our approach leads to a direct and compact solution of this problem.

Cell-to-cell heterogeneity drives a range of (patho)physiologically important phenomena, such as cell fate and chemotherapeutic resistance. The role of metabolism, and particularly mitochondria, is increasingly being recognised as an important explanatory factor in cell-to-cell heterogeneity. Most eukaryotic cells possess a population of mitochondria, in the sense that mitochondrial DNA (mtDNA) is held in multiple copies per cell, where the sequence of each molecule can vary. Hence intra-cellular mitochondrial heterogeneity is possible, which can induce inter-cellular mitochondrial heterogeneity, and may drive aspects of cellular noise. In this review, we discuss sources of mitochondrial heterogeneity (variations between mitochondria in the same cell, and mitochondrial variations between supposedly identical cells) from both genetic and non-genetic perspectives, and mitochondrial genotype-phenotype links. We discuss the apparent homeostasis of mtDNA copy number, the observation of pervasive intra-cellular mtDNA mutation (we term `microheteroplasmy') and developments in the understanding of inter-cellular mtDNA mutation (`macroheteroplasmy'). We point to the relationship between mitochondrial supercomplexes, cristal structure, pH and cardiolipin as a potential amplifier of the mitochondrial genotype-phenotype link. We also discuss mitochondrial membrane potential and networks as sources of mitochondrial heterogeneity, and their influence upon the mitochondrial genome. Finally, we revisit the idea of mitochondrial complementation as a means of dampening mitochondrial genotype-phenotype links in light of recent experimental developments. The diverse sources of mitochondrial heterogeneity, as well as their increasingly recognised role in contributing to cellular heterogeneity, highlights the need for future single-cell mitochondrial measurements in the context of cellular noise studies.

Mitochondrial DNA (mtDNA) mutations cause severe congenital diseases but may also be associated with healthy aging. MtDNA is stochastically replicated and degraded, and exists within organelles which undergo dynamic fusion and fission. The role of the resulting mitochondrial networks in the time evolution of the cellular proportion of mutated mtDNA molecules (heteroplasmy), and cell-to-cell variability in heteroplasmy (heteroplasmy variance), remains incompletely understood. Heteroplasmy variance is particularly important since it modulates the number of pathological cells in a tissue. Here, we provide the first wide-reaching theoretical framework which bridges mitochondrial network and genetic states. We show that, under a range of conditions, the (genetic) rate of increase in heteroplasmy variance and de novo mutation are proportionally modulated by the (physical) fraction of unfused mitochondria, independently of the absolute fission-fusion rate. In the context of selective fusion, we show that intermediate fusion/fission ratios are optimal for the clearance of mtDNA mutants. Our findings imply that modulating network state, mitophagy rate and copy number to slow down heteroplasmy dynamics when mean heteroplasmy is low could have therapeutic advantages for mitochondrial disease and healthy aging.

It has been suggested that microtubules and other cytoskeletal filaments may act as electrical transmission lines. An electrical circuit model of the microtubule is constructed incorporating features of its cylindrical structure with nanopores in its walls. This model is used to study how ionic conductance along the lumen is affected by flux through the nanopores when an external potential is applied across its two ends. Based on the results of Brownian dynamics simulations, the nanopores were found to have asymmetric inner and outer conductances, manifested as nonlinear IV curves. Our simulations indicate that a combination of this asymmetry and an internal voltage source arising from the motion of the C-terminal tails causes a net current to be pumped across the microtubule wall and propagate down the microtubule through the lumen. This effect is demonstrated to enhance and add directly to the longitudinal current through the lumen resulting from an external voltage source, and could be significant in amplifying low-intensity endogenous currents within the cellular environment or as a nano-bioelectronic device.

Protein molecules in cells are synthesized by macromolecular machines called ribosomes. According to recent experimental data, we reduce the complexity of the ribosome and propose a model to express its activity in six main states. Using our model, we study the translation rate in different biological relevant situations in the presence of external force, and translation through the RNA double stranded region in the absence or presence of the external force. In the present study, we give a quantitative theory for translation rate and show that the ribosome behaves more like a Brownian Ratchet motor. Our findings could shed some light on understanding behaviors of the ribosome in biological conditions.