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Abstract The last decade has witnessed a surge of theoretical and computational models to describe the dynamics of complex gene regulatory networks, and how these interactions can give rise to multistable and heterogeneous cell populations. As the use of theoretical modeling to describe genetic and biochemical circuits becomes more widespread, theoreticians with mathematical and physical backgrounds routinely apply concepts from statistical physics, non-linear dynamics, and network theory to biological systems. This review aims at providing a clear overview of the most important methodologies applied in the field while highlighting current and future challenges. It also includes hands-on tutorials to solve and simulate some of the archetypical biological system models used in the field. Furthermore, we provide concrete examples from the existing literature for theoreticians that wish to explore this fast-developing field. Whenever possible, we highlight the similarities and differences between biochemical and regulatory networks and ‘classical’ systems typically studied in non-equilibrium statistical and quantum mechanics. 

Understanding the mechanisms governing the structure and dynamics of flexible polymers like chromosomes, especially the signatures of motor-driven active processes, is of great interest in genome biology. We study chromosomes as a coarse-grained polymer model where microscopic motor activity is captured via an additive temporally persistent noise. The active steady state is characterized by two parameters: active force, controlling the persistent-noise amplitude, and correlation time, the decay time of active noise. We find that activity drives correlated motion over long distances and a regime of dynamic compaction into a globally collapsed entangled globule. Diminished topological constraints destabilize the entangled globule, and the active segments trapped in the globule move toward the periphery, resulting in an enriched active monomer density near the periphery. We also show that heterogeneous activity leads to the segregation of the highly dynamic species from the less dynamic one, suggesting a role of activity in chromosome compartmental segregation. Adding activity to experimental-data-derived structures, we find active loci may mechanically perturb and switch compartments established via epigenetics-driven passive self-association. The key distinguishing signatures of activity are enhanced apparent diffusivity, exploration of all the dynamic regimes (subdiffusion, effective diffusion, and superdiffusion) at various lag times, and a broadened distribution of observables like the dynamic exponents. Published by the American Physical Society2024 

High-resolution techniques capable of manipulating from single molecules to millions of cells are combined with three-dimensional modeling followed by simulation to comprehend the specific aspects of chromosomes. From the theoretical perspective, the energy landscape theory from protein folding inspired the development of the minimal chromatin model (MiChroM). In this work, two biologically relevant MiChroM energy terms were minimized under different conditions, revealing a competition between loci compartmentalization and motor-driven activity mechanisms in chromatin folding. Enhancing the motor activity energy baseline increased the lengthwise compaction and reduced the polymer entanglement. Concomitantly, decreasing compartmentalization-related interactions reduced the overall polymer collapse, although compartmentalization given by the microphase separation remained almost intact. For multiple chromosome simulations, increased motorization intensified the territory formation of the different chains and reduced compartmentalization strength lowered the probability of contact formation of different loci between multiple chains, approximating to the experimental inter-contacts of the human chromosomes. These findings have direct implications for experimental data-driven chromosome modeling, specially those involving multiple chromosomes. The interplay between phase-separation and territory formation mechanisms should be properly implemented in order to recover the genome architecture and dynamics, features that might play critical roles in regulating nuclear functions. 

During mitosis, there are significant structural changes in chromosomes. We used a maximum entropy approach to invert experimental Hi-C data to generate effective energy landscapes for chromosomal structures at different stages during the cell cycle. Modeled mitotic structures show a hierarchical organization of helices of helices. High-periodicity loops span hundreds of kilobases or less, while the other low-periodicity ones are larger in genomic separation, spanning several megabases. The structural ensembles reveal a progressive decrease in compartmentalization from interphase to mitosis, accompanied by the appearance of a second diagonal in prometaphase, indicating an organized array of loops. While there is a local tendency to form chiral helices, overall, no preferential left-handed or right-handed chirality appears to develop on the time scale of the cell cycle. Chromatin thus appears to be a liquid crystal containing numerous defects that anneal rather slowly. 

Recent advancements in RNA structural biology have focused on unraveling the complexities of non-coding mRNA elements like riboswitches. These cis-acting regulatory regions undergo structural changes in response to specific cellular metabolites, leading to up or downregulation of downstream genes. The purine riboswitch family regulates many prokaryotic genes involved in purine degradation and biosynthesis. They feature an aptamer domain organized around a 3-way helical junction, where ligand encapsulation occurs at the junctional core. In our study, we chemically probed the aptamer domain of the 2’-dG-sensing purine riboswitch from Mesoplasma florum (dGsw) under various solution conditions to understand how Mg²⁺ and 2’-dG influence riboswitch folding. Here, we find that efficient 2’-dG binding strongly depends on Mg²⁺, indicating that Mg²⁺ is essential for priming dGsw for ligand interactions. We identified a previously undescribed sequence in the 5’ tail of dGsw that is complementary to a conserved helix. The inclusion of this region in a construct led to intramolecular competition between the alternate helix, Palt, and P1. Mutational analysis confirmed that 5’ flanking end of the aptamer domain forms an alternate helix in the absence of ligand. Molecular dynamics simulations revealed that this alternative conformation is stable. This helix may, therefore, facilitate the formation of an anti-terminator helix by opening the 3-way junction surrounding the 2’-dG binding site. Our study further establishes the importance of a closed terminal P1 helix conformation for metabolite binding and suggests that the delicate interplay between P1 and Palt may fine-tune downstream gene regulation. These insights offer a new perspective on riboswitch structure and enhance our understanding of the role that a conformational ensemble plays in riboswitch activity and regulation. 

Electron transfer is at the heart of many fundamental physical, chemical, and biochemical processes essential for life. The exact simulation of these reactions is often hindered by the large number of degrees of freedom and by the essential role of quantum effects. Here, we experimentally simulate a paradigmatic model of molecular electron transfer using a multispecies trapped-ion crystal, where the donor-acceptor gap, the electronic and vibronic couplings, and the bath relaxation dynamics can all be controlled independently. By manipulating both the ground-state and optical qubits, we observe the real-time dynamics of the spin excitation, measuring the transfer rate in several regimes of adiabaticity and relaxation dynamics. Our results provide a testing ground for increasingly rich models of molecular excitation transfer processes that are relevant for molecular electronics and light-harvesting systems. 

This study presents an enhanced protein design algorithm that aims to emulate natural heterogeneity of protein sequences. Initial analysis revealed that natural proteins exhibit a permutation composition lower than the theoretical maximum, suggesting a selective utilization of the 20-letter amino acid alphabet. By not constraining the amino acid composition of the protein sequence but instead allowing random reshuffling of the composition, the resulting design algorithm generates sequences that maintain lower permutation compositions in equilibrium, aligning closely with natural proteins. Folding free energy computations demonstrated that the designed sequences refold to their native structures with high precision, except for proteins with large disordered regions. In addition, direct coupling analysis showed a strong correlation between predicted and actual protein contacts, with accuracy exceeding 82% for a large number of top pairs (>4L). The algorithm also resolved biases in previous designs, ensuring a more accurate representation of protein interactions. Overall, it not only mimics the natural heterogeneity of proteins but also ensures correct folding, marking a significant advancement in protein design and engineering.