A Typical Workflow to Simulate Cytoskeletal Systems

The cytoskeleton consists of filaments within the cell and associated molecules such as molecular motors, which often constitute a dynamic meshwork with remarkable mechanical properties. The cytoskeleton exists in various configurations in different cell types across nearly all life forms. Its correct functioning is essential for fundamental cellular processes such as division, motility, and polarization. It also governs cell-to-cell mechanical interactions, thereby influencing the morphogenesis of tissues and organisms. The cytoskeleton underlies several functions and manifests itself in many biological processes. For example, the contraction of muscles is linked to the power stroke of myosin molecular motors on actin filaments. Another example is the maintenance of neurons, which relies on the movements of kinesin motors along microtubules located inside the axons of these neurons. Actin and microtubules are two preeminent types of cytoskeletal filaments, and without them, life as we know it would be impossible.

The cytoskeleton is essentially a biomechanical system, which cannot be reduced solely to its chemistry. Microtubules or actin filaments are built from thousands of monomers and extend over several micrometers. The conformations of these filaments in space and the forces that they can transmit to the plasma membrane, the nucleus, or other organelles are key aspects of their role in the cell. For example, a network of actin filaments and myosin motors, called the actomyosin cortex¹, generates forces to sustain cell motility and morphological changes in animal cells. A very different arrangement is seen in plant cells, where cortical microtubules direct the deposition of cellulose fibrils, thereby controlling the cell wall architecture, which ultimately determines how these cells will grow in the future².

While mechanics clearly play a substantial part in cytoskeletal operations, chemistry is equally important. Filaments grow via a self-assembly process whereby monomers find their docking site at the tip of the filament after diffusing through the cytoplasm³. At the molecular scale, assembly and disassembly at the tip of the filaments are, thus, determined by molecular affinities⁴. Similarly, proteins of the cytoskeleton diffuse, and binding and unbinding rates determine their affinity for the filaments they encounter. In the case of molecular motors, cycles of chemical reactions involving ATP hydrolysis are linked to movements along the filaments and, possibly, forces that accompany them⁵. Remarkably, the cytoskeleton offers many unusual challenges and a large variety of processes involving similar components. It is a rich playground at the interface between biology, chemistry, and physics.

Cytoskeletal systems are amenable to mathematical modeling. In fact, thanks to excellent research done in the past decades, the principal molecular constituents are most likely already identified, as illustrated with endocytosis⁶. In model organisms, such as yeast, the properties of these elements are known, as well as the system composition for some of their processes. For instance, the structure and material properties of microtubules⁷, as well as their number and average lengths at various stages of the mitotic spindle, have been described⁸. The number of kinesins that connect microtubules into a coherent mechanical structure is often known⁹. The speeds of many motors have been measured in vitro¹⁰. In addition, experimentalists can observe and quantify these systems in vivo under wild-type or mutated conditions. Combining theory alongside in vivo and in vitro experiments enables researchers to test whether the current knowledge about a cytoskeletal system is sufficient to explain its observed behavior. The use of mathematical and computational tools also allows us to make inferences of how components work collectively on the basis of assumptions derived from observations at the molecular scale, usually in simplified situations (e.g., single-molecule experiments).

The role of theory can be illustrated using a practical example: the beating of cilia. This beating is due to the movement of dynein motors along microtubules in the cilia. One may ask what determines the speed of the dynein motor in this system. One possible answer is that the maximum speed is constrained by the requirement to maintain a certain beating pattern. This would be understandable if the beating was under natural selection. In that case, if motors moved quicker, then the process would lose its desired qualities-the cilia would not beat as efficiently or even fail altogether. Although this is possible, a second alternative is that some intrinsic factor could limit dynein's speed.

For example, the cell may not have enough ATP to make dynein faster, or the protein movements required for dynein's activity just could not be accelerated. In that case, if the motors could be made faster despite the physical limits, the beating would be improved. A third possibility, of course, is that changing the speed does not affect the process significantly, which might be advantageous to the organism by providing some "robustness" against uncontrollable factors. Amongst these three possibilities, one can identify the correct one by calculating the beating pattern from dynein's properties. Indeed, a suitable mathematical model should predict how the beating pattern is affected by varying dynein's speed and is not subject to the limits that exist in the physical world. Naturally, the validity of the model must be verified, but even "incorrect" models can generate interesting ideas.

The model can take the form of an analytical framework or be a numerical simulation of the system. Either way, the gap between the molecular scale and the functional scale remains an obstacle, and developing these models is not a straightforward task, since several mechanical and chemical processes need to be integrated into the equations describing the biological system. Theory comes in various forms, offering different tradeoffs between simplicity and realism. Increasing the degree of details in a model is not always advantageous as it may limit our ability to solve the equations or, in other words, to derive the predictions of the theory. The same tradeoff exists for simulations. The modelers will have to select the ingredients of the system to be considered while ignoring certain aspects. These key decisions will depend strongly on the objective of the study. Nowadays, the extraordinary improvements in computer hardware make it possible to simulate many cytoskeletal systems with enough details over a sufficient time to analyze their comportment. This will often generate unexpected ideas and novel directions in research. For example, simulations similar to the ones that will be used in this protocol led to a back-of-the-envelope calculation that can predict the contractility of a network based on its composition¹¹.

Numerical methods are ubiquitous in engineering and physical sciences, and their use in biology is growing. Today, virtually all our technological whatchamacallit (watches, phones, cars, and computers) has been first conceived on a computer, and powerful software exist to do this. Given a well-characterized cytoskeletal system and assuming that an appropriate level of description has been determined, several issues must still be solved before it can be simulated. For the simplest problems, the most appropriate route of action might be to write a simulation "by coding from scratch", in other words, starting with a generic programming language or a mathematical platform such as MATLAB. This has the advantage that the author of the code will have an intimate knowledge of what has been implemented and knows exactly how the software works. This route is, however, not without risk, and it is not uncommon to witness doctoral students spending most of their working time writing code rather than addressing scientific questions.

The alternative is to use software conceived by others, but this is not without risks either; any large source code tends to spontaneously acquire the traits of an impenetrable black box, despite the most admirable efforts of their authors to prevent it. Using black boxes is surely not a scientist's dream. A large source code can also become a liability, and it may be quicker to start from scratch than to modify an existing code base to make it do something different. To mitigate this problem, one can always invite the authors of the software to help, but this may not be sufficient. Frequently, there is a difference of scientific culture between the authors of the software and the people that would like to use it, which means that many implicit assumptions need to be clarified. By making the code Open Source, it is expected that more people will be involved in the development of the software and to maintain its documentation, thus improving its quality. All these are important issues that must be given proper consideration before any investment is made. Nevertheless, the only way to progress in the long term is to promote solid software solutions, used and maintained by a broad community with common scientific interests.

Although this protocol uses Cytosim, there are other Open Source tools that might be able to simulate the same system, for example, AFINES¹², MEDYAN¹³, CyLaKS¹⁴, aLENS¹⁵, and AKYT¹⁶, to name a few. Unfortunately, comparing these projects is beyond the scope of the article. Here, step-by-step instructions are given to simulate a contractile 2D actomyosin network. This system is simple and makes use of the better-established capacities of Cytosim. Cytosim is built around a cross-platform core engine that can run simulations in 2D or 3D. It has a modular code base, making it easily customizable to perform particular tasks. Cytosim is equally stable and efficient in 3D and has been successfully used in the past to investigate diverse problems involving microtubules and actin filaments: the association of two asters of microtubules¹⁷, the movement of nuclei in the cells^18,19, endocytosis⁶, cytokinesis²⁰, the formation of the mitotic spindle²¹, the movements of the mitotic spindle²², the capture of chromosomes²³, the contraction of actomyosin networks^11,24, and the mechanics of the microtubule ring in blood platelets²⁵, and the capacities developed for these projects have been maintained in the code. The workflow described here can be adapted to many other problems. It makes use of the Unix command line, which may be unfamiliar to some readers. Using the command line is, however, the most portable and convenient way to automate the process of running simulations. Integrated graphical user interfaces aim to offer easy, intuitive access to a software, but this often comes at the expense of generality. The objective of this article is to illustrate an approach that can easily be modified or adapted to other problems. Notes are provided to explain the meaning of the commands.

To simulate an actomyosin network, filaments are modeled as oriented lines and represented by vertices distributed along their length (Figure 1). This is an intermediate level of description, common in polymer physics, that ignores the genuine 3D nature of the filaments but allows the bending to be calculated. Filaments may grow and shrink at their ends, following different models that cover both actin and microtubule phenomenology. In cells, filaments are organized primarily through interactions that constrain their motion, for example, the attachment to other filaments or simply confinement within the cell. In Cytosim, all such interactions are linearized and combined in a large matrix²⁶. The equations describing the motion of all filament vertices are derived from this matrix, assuming a viscous medium and random fluctuating terms representing Brownian motion. These equations are solved numerically to obtain the motion of the filaments together with all the forces acting on them in a self-consistent and efficient manner²⁶. Superimposed on this mechanical engine, there is a stochastic engine that simulates discrete events, such as the attachments and detachments of molecular motors or the assembly dynamics of filaments. In summary, Cytosim first uses simulated dynamics to calculate the mechanics of a network of filaments, connected in any arbitrary manner, and, second, stochastic methods to simulate the binding, unbinding, and diffusion of proteins that connect or affect the filaments.

The workflow illustrated here was frequently followed to initially explore a system using Cytosim. The critical step for many potential users is likely to be the installation of the software components. Distributing the software as a source code fulfills the imperatives of Open Science, but it is prone to errors since the developers of the software have access to only a limited pool of architecture to test the program. Compilation may fail as operating systems differ. The instructions provided here are likely to become obsolete as computer systems and source codes evolve. Thus, periodically checking the latest instructions online is essential. In case of trouble, it is highly encouraged that users report back by posting on the relevant feedback channel (currently Cytosim's homepage on Gitlab) to help fix the problem.