Physical Modeling of microtubule force generation and self-organization

Biological systems are complex heterogeneous and far from equilibrium systems. The fundamental questions posed by the physics of such systems are what the force generation mechanisms are, and how energy is processed and distributed among the components inside them. In answering these questions we can understand how motion is generated and how the system is organized, which means a significant step toward grasping these systems in their full complexity. A systematic program means first the identification of the components, and studying its properties and interplay with other components. How these components integrates into a higher level of organization, comes as a secondary step.

The cytoskeleton is a key ingredient of the living cell. The cytoskeleton is a complex of biopoly-mers which self-assembles and organize inside the living cells. There are many important functions that cytoskeleton fulfills. One is to give shape and rigidity to the cell, another is that cytoskeletal biopolymers serves as tracks for material transport across the cell. The examples could continue with the locomotion of cell, which is possible only due to the rearrangement of the cytoskeleton.

This thesis is concerned with the physical aspects of microtubules, which represent a part of the cytoskeleton. Microtubules are tubular protein aggregates, which are particularly stiff. These biopolymers were originally discovered as the scaffold of the mitotic spindle, which is the cell division apparatus that separates the genetic material among the daughter cells. An important property of microtubules is the alternation between growing and shrinking states, a behavior which is termed as dynamic instability and make microtubules unique in the realm of polymers. It is precisely this property that makes possible for microtubules to be involved in multi scale dynamics, i.e. assembly-disassembly and organization.

In making the time scale separation, some particular aspects of microtubule assembly and organization are presented and analyzed in two different parts of this thesis. The attention is focused on growing microtubules only, i.e. the dynamic instability does not play any role in the processes that are considered.

In the first part of this thesis, it is investigated in detail the microtubule force production mechanism during self-assembly. In general, any polymer can generate force during polymerization. If the seed of the polymer is fixed, then polymer can push against an arbitrary object, if the insertion of subunits are allowed due to gap opening between the tip of the polymer and the corresponding object. The required gap openings are possible due to the thermal fluctuations, and it is due to this reason that the object that generates force by exploiting the thermal fluctuations is called Brownian ratchet. This particular type of motor does not contradicts the second law of thermodynamics, which forbids work production in isothermal systems. The problem is avoided as the system is out of equilibrium. In our example of the polymerization ratchet, the dynamics is driven by the chemical polymerization energy, which is simply converted into work by the Brownian ratchet mechanism itself. Microtubules that work as Brownian ratchets can be regarded as a particular type of a molecular (nano-)motor.

In Chapter 3, the concept of Brownian ratchet is applied to microtubules. The main feature which is incorporated to this concept is the collective character of the microtubule growth, since these polymers are composed of many filaments. One important question is to investigate what is the maximum force that this particular type of molecular motor can generate. A second question is to see how the velocity of growth depends on the opposing force that an external object can exert. Does the velocity of growth depend on the relative arrangements of microtubule protofilaments inside the assembly? In other words, given its internal structure is there a optimal way that the microtubule can grow under load condition? The way that the investigation is carried out is that the model details are extracted with the help of computer simulations, and compared directly with experimental data.

In Chapter 4, different regimes of microtubule growth are considered. Quantitative comparisons with available experimental data are successful in all cases, but a large number of free parameters justifies the need for different experiments. However, some qualitative aspects, such as the microtubule end structure can limit the number of possibilities, since end details were already observed in experiments. More exactly, cryo-electron microscope images show that microtubuls develop open sheets like structures at their end during growth. The disappearance of these structures is correlated in experiments with a hypothetical switch mechanism that triggers dynamic instabilities. Therefore, it appears natural to expect that a realistic growth model should reproduce such end structures. The model suggests that there is a sensitive relationship between the size of these structures and both the kinetic rates and the strength of the lateral bonds between protofilaments. Although the comparison with experiments is not fully quantitative, the analysis suggests that it is likely that the lateral bonding between the protofilaments is relatively week, Le. a couple of thermal energies kg T per subunit.

In the second part of the thesis, I discuss some physical aspects regarding the organization of microtubules. In general, not referring only to microtubules, the importance of understanding the cytoskeleton organization is manifold. From physical point of view, the questions that are addressed in this thesis belong to the much broader context of pattern formation in far from equilibrium systems. Here, the fundamental problem is to find the relationship between the macroscopic properties of organized dissipative systems and their microscopic details that drive the sys-tem out of thermodynamic equilibrium. From the biological point of view, the investigation of the cytoskeleton organization is tightly related to understanding the biological functional role that different biopolymer arrangements assume in living cells.

In Chapter 5, the attention is focused on the microtubule organization in higher plant cells. Particularly, the microtubule arrangements that appear in interphase cells or prior to their division have received a lot of attention from biologists in the past, but still little is known about the driving organization mechanism. In interphase plant cells, the microtubules organize on the cortex of the cell in a parallel array, which is oriented transversely to the main axis of the cell. Just before the onset of the division, this array narrows to a preprophase band which marks on the cortex the location of the separation wall between the daughter cells. From physical point of view, in this chapter is addressed the question if it is possible that passive factors could be responsible for such organized arrays. One possibility in this respect is the nematic transition driven by excluded volume interaction, which is a well known phenomenology from the physics of liquid crystals. This implies a direct relationship between the degree of ordering and the density of microtubules. A second possibility is that bending elasticity of microtubules is the driving factor for organizing microtubules on the cortex. Since the bending elasticity is an intrinsic property of microtubules, the organization in this case can be termed more exactly as se/f-organization.

Active factors are the best candidates in driving large scale patterns in filamentous systems. In the past, the ability of motor proteins to organize filaments is demonstrated in both experiments and computer simulations. However, understanding the phase diagram remains an open theoretical problem. Based on phase diagram analysis, a minimal set of conditions can be derived in order to reproduce a particular phenomenology. In the last two chapters, two different approaches are adopted. In Chapter 6, a mean field Landau type theory is developed. In this case, the phenomenology of filamentous systems is described with no reference to microscopic details and the basic constrains, which are imposed, are the symmetries that the physical system is supposed to fulfill. This generic method reproduce the possibility of a transverse stripe that closely resembles the preprophase band in plant cells. This encouraging result suggests that cytoskeletal array like those observed in plant cells can be described by a mean field theory.

In Chapter 7, a microscopic model is introduced, and based on this I derive the macroscopic evolution equations. The procedure is meant to meet the results that are derived in the generic approach, which is presented in Chapter 6, Besides the active components, I introduced also the passive interaction due to steric exclusion between filaments. The passive components alone are responsible for isotropic-nematic instabilities at high density, which drive the system to a liquid crystalline ordered phase. However, the active components can drive pattern formation in this system at densities that are below the critical value that corresponds to passive driven instabilities. The study in this chapter is limited at the level of linear stability analysis. However, the obtained results suggest that the stable arrays might be homogeneous nematic polar patterns, vortices, and asters. These features are consistent with the results obtained from other methods, like computer simulations or in vitro experiments, which are present in the literature. A full understanding of the emergent patterns requires the consideration of non-linear terms in evolution equations, which is the objective of future projects.