6 Conclusions, applications, and perspectives
6.1 Summary of the main contributions
This manuscript has highlighed our main contributions to the research on the modeling of muscle contraction at different scales. At the nanoscale (Chapter 3), we have formulated a stochastic jump-diffusion model of actin-myosin interaction, combining a continuous description of the power-stroke conformational change with a mode classical discrete representation of the attachment-detachment process. In particular, we have shown that this extension of the classical Huxley-Hill framework can be made compatible with the thermodynamics principle provided an adequate detailed balance condition. This model can be robustly calibrated to reproduce state of the art mechanical experiments performed on muscle fibers.
At the microscale (Chapter 4), we investigated the role played by elastic interactions in the synchronisation of the cross-bridges conformational changes. The strength of these elastic interactions competes with the thermal agitation, which generate phase transitions analog to the ferromagnetic to paramagnetic transition of the Curie-Weiss model. We discuss the implication of such synchronisation on the behavior of sarcomeres submitted to fast load changes, which we show to be ensemble-dependent. We presented a direct applciation of this work where we show the effect of long-range mechanical interaction on the fine tuning of the SNARE machinery in the process of neurotransmitter release.
Linking the nanoscale to the macroscale within a genuinely multiscale framework is where major efforts remain to be made (Chapter 5). Recent experimental studies have shed light on the essential role played by structural proteins in regulation processes involving long-range mechanical feedback loops. The nature of the internal elastic coupling can make difficult to define adequate “small parameters” necessary for applying standard upscaling techniques. A fully detailed 3D representation of the sarcomeric internal structure is possible but at a very high numerical cost, so new reduced models will have to be developed.
6.2 Applications
In this section, we present potential applications of our work.
6.2.1 Collective switching of systems with long-range interactions
An extended version of this discussion can be found in ref. (Caruel and Truskinovsky 2018, sec. 6).
The prototypical nature of the main model formulated for the study of the collective conformational change among an ensemble of interconnected cross-bridges (see Chapter 4), makes it relevant to beyond the muscle context. It describes a molecular device capable of converting a continuous input (in the form of an external load change in the case of muscles) into an all-or-none output. Such conversion mechanism is ubiquitous in cellular physiology.
A first example is the cooperative zippering of SNAREpins presented in Section 4.3.7. Another example is provided by the transduction channels in the hair cells bundle of the auditory system. In these bundles, the motion induced by the sound waves triger the opening and closing of elastic ion gates in response to the deformation of the cilia of the cells, see (Bormuth et al. 2014). A last example is the analogy that can be drawn between the cross-bridge model and the models of collective unzipping for adhesive clusters containing bundles of bistable proteins, such as integrins or cadherins, connected to elastic substrates (Erdmann and Schwarz 2007; Yao and Gao 2006; Gao, Qian, and Chen 2011).
Furthermore, collective conformational changes in distributed biological systems containing coupled bistable units can be driven not only mechanically, by applying forces or displacements, but also biochemically by, varying concentrations or chemical potentials of ligand molecules in the environment (Changeux et al. 1967; Monod, Wyman, and Changeux 1965). The response of these so-called allosteric systems is ultrasensitive to changes in the ligand concentration, leading to all-or-none type of responses. The long range coupling is often provided by mechanical stresses inside membranes and macromolecular complexes,. Notice that the actin filament activation mentioned in Section 5.1.4 belongs to this class of system.
Finally, our work on the cooperative power-stroke within contractile units has been used for the design of mechanical devices taking advantage of engineered long-range internal interactions, see (Harne, Wu, and Wang 2016; Wu, Harne, and Wang 2016; Harne, Wu, and Wang 2015; Kidambi, Harne, and Wang 2017). In these work, prototypes of modular metastructures are designed, showing exploitable metastable states and adjustable hysteresis. This type of structure can be used for instance for energy harvesting based on asymetric design of the bistable elements, or for the control of wave propagation if the system, in the case of series connexion (Nadkarni et al. 2016).
6.2.2 Simulation of muscle contraction
The benefits of using of multiscale models of biological tissue in clinical context are well identified. They are directly connected to the development of “numerical twins” to test surgical and monitoring proceedures or drugs. One aim of modeling efforts is thus to provide tools that would be used routinely in the clinic, or tools that would be used for treatment design or clinical research purposes.
As an example of the first use case, we can highlight the start-up led by François Kimmig. This company, named AnaestAssist, is the result of a collaboration between the Inria MΞDISIM team and the Anesthesia and Intensive Care unit at Lariboisière Hospital (APHP). AnaestAssist aims to provide an enhanced monitoring solution integrated with an alert and decision support tool. By utilizing real-time simulations that interact with data, AnaestAssist predicts the cardiovascular response to anesthetic pharmacology during surgery, thereby reducing the risk of hypotension.
To illustrate the second use case, we can mention the work of Pr Linari’s group (PhysioLab, University of Florence), who is involved in a synergistic european research project with clinicians on investigating the role of titin in skeletal muscle disorders and trying to correlate the phenotype to the genotype. One aspect of the methodology consists in testing samples from patients to measure the functional consequences of the genetic variations and finally improve the clinical diagnostic pipeline for skeletal muscle disorder. Modeling could be used to supplement this kind of procedure with simulations to bring additional mechanistic insight into clinical diagnostic and decision in the context of genetic cardiomyopathies.
A practical example for pharmaceutical development is to predict the effect of isolated using intense in silico screaning on the tissue behavior using multiscale modeling. This modeling approach can be supplemented by experiments using reconstructed biomimetic systems using, for instance the nanomachine developped at the PhysioLab by the group of P. Bianco, see Section 4.1.1.
6.3 Perspective and future work
To pursue our research we identified the following projects.
6.3.1 Mesoscale modeling
As we have shown in @#sec-chapter-mesoscale, one of the biggest challenge currently addressed by physiologist and biophysicists working on muscle contraction is the understanding of the regulation mechanisms of contraction that involves the structural proteins connecting contractile units together.
To contribute to this effort, we aim at formulating a model of the myofilament lattive using effective contractile unit models. The reduced models should be obtained using adequate asymptotic representation of a fully explicit contractile unit stochastic model with a finite number of motors. To achieve this step we started a collaboration with the CERMICS laboratory (Center for Training and Research in MathematIcs and Scientific Computing) at Ecole des Ponts, in particular with Pr. Tony Lelièvre and Pr. Julien Reygnier. The question addressed are the compatibility of this effective model with in-situ observations of the average molecular motors conformations in muscle fibers (in situ X-ray experiments), and the validity of the common mean-field approach.
A possible next step for that project is to devformulate a homogenized model law of the mesoscale contractile unit network by considering appropriate continuum limits in the direction of the fiber. The ensuing equations characterizing the level of deformation and the corresponding force will be coupled to the continuum mechanics balance laws within the framework presented in Chapter 2. We will develop the numerical methods to implement the new multiscale contraction model in the existing heart simulation framework developed over the past years with the MΞDISIM Inria team.
6.3.2 Tissue engineering
A project we intend to pursue consists in determining the effects of the mesoscale structure on the mechanical output of bioprinted tissue. Bioprinting technologies have made tremendous progress in the past decade, to the point where it becomes feasible to test treatments on bioprinted samples and repair injured tissue (Ma et al. 2023). When addressing the issue of modeling muscle contraction at the mesoscale, using bioprinted tissue presents several against classical approaches based on dissected tissues:
- Cells can be grown within a few days from standardized cell lines, which is simpler that using actual animal models.
- Many cells can be observed at once which means that some experiments can be parallelized.
- The bioprinter allow a good control of the microstructure: we could produce tissues with various distribution of fiber orientations for instance.
- The experiments may be more reproducible.
Such a project could be decomposed into four work packages:
- Develop the bioprinting protocol. The objective is here to obtain a contracting tissue with desired microstructure
- Define the readouts. We intend to measure the deformation field of the tissue and link this deformation to the microstructure. The deformation can be measured for instance using microbeads printed with the biomaterial. Microstructure can be characterized using standard microscopy techniques, like Two Photon Electro Fluorescence. Intracellular component are typically observed using live stain imaging. Finally, the morphology of the tissue can be characterized using Atomic Force Microscopy, potentially in real time.
- Design adequate loading devices. Mechanical tests on muscle fibers are performed only in 1D, in the direction of the fiber. However, in vivo, the tissue is subjected to more complex loaddings, which we aim to reproduce in vitro. To reach this objective, the simplest way is to design first a biaxial traction machine. We can also use 3D fabricated surfaces covered with micropillar and measure local force by recording the mucropillar deflexion. Traction Force microscopy is another solution.
- Formulate new models. Here a first step is to improve the existing model to take into accound fiber anisotropy in an already homogenized framework. The homogenization of a single fiber can then be adressed to determine if developed heterogeneities in the sarcomere deformation has a macroscopic signature. At a smaller scale, we can also propose more descriptive models of how mechanical forces can influence muscle activation and regulation physiological mechanisms, as mentioned previously.