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There are innumerable 'biological complexity measure's. While some patterns emerge from these attempts to represent biological complexity, a single measure to encompass the seemingly countless features of biological systems, still eludes the students of Biology. It is the pursuit of this paper to discuss the feasibility of finding one complete and objective measure for biological complexity. A theoretical construct (the 'Thread-Mesh model') is proposed here to describe biological reality. It segments the entire biological space-time in a series of different biological organizations before modeling the property space of each of these organizations with computational and topological constructs. Acknowledging emergence as a key biological property, it has been proved here that the quest for an objective and all-encompassing biological complexity measure would necessarily end up in failure. Since any study of biological complexity is rooted in the knowledge of biological reality, an expression for possible limit of human knowledge about ontological biological reality, in the form of an uncertainty principle, is proposed here. Two theorems are proposed to model the fundamental limitatio

Extreme value analysis (EVA) is a statistical method that studies the properties of extreme values of datasets, crucial for fields like engineering, meteorology, finance, insurance, and environmental science. EVA models extreme events using distributions such as Fréchet, Weibull, or Gumbel, aiding in risk prediction and management. This review explores EVA's application to nanoscale biological systems. Traditionally, biological research focuses on average values from repeated experiments. However, EVA offers insights into molecular mechanisms by examining extreme data points. We introduce EVA's concepts with simulations and review its use in studying motor protein movements within cells, highlighting the importance of in vivo analysis due to the complex intracellular environment. We suggest EVA as a tool for extracting motor proteins' physical properties in vivo and discuss its potential in other biological systems. While there have been only a few applications of EVA to biological systems, it holds promise for uncovering hidden properties in extreme data, promoting its broader application in life sciences.

Many biological processes can be thought of as the result of an underlying dynamics in which the system repeatedly undergoes distinct and abortive trajectories with the dynamical process only ending when some specific process, purpose, structure or function is achieved. A classic example is the way in which microtubules attach to kinetochores as a prerequisite for chromosome segregation and cell division. In this example, the dynamics is characterized by apparently futile time histories in which microtubules repeatedly grow and shrink without chromosomal attachment. We hypothesize that for biological processes for which it is not the initial conditions that matter, but rather the final state, this kind of exploratory dynamics is biology's unique and necessary solution to achieving these functions with high fidelity. This kind of cause and effect relationship can be contrasted to examples from physics and chemistry where the initial conditions determine the outcome. In this paper, we examine the similarities of many biological processes that depend upon random trajectories starting from the initial state and the selection of subsets of these trajectories to achieve some desired func

Morphogenesis, the process of growth and shape formation in biological tissues, is driven by complex interactions between mechanical, biochemical, and genetic factors. Traditional models of biological growth often rely on the concept of homeostatic Eshelby stress, which defines an ideal target state for the growing body. Any local deviation from this state triggers growth and remodelling, aimed at restoring balance between mechanical forces and biological adaptation. Despite its relevance in the biomechanical context, the nature of homeostatic stress remains elusive, with its value and spatial distribution often chosen arbitrarily, lacking a clear biological interpretation or understanding of its connection to the lower scales of the tissue. To bring clarity on the nature of homeostatic stress, we shift the focus from Eshelby stress to growth incompatibility, a measure of geometric frustration in the tissue that is the primary source of residual stresses in the developing body. Incompatibility, measured by the Ricci tensor of the growth metric at the continuous level, can be potentially regulated at the cell level through connections with the surrounding cells, making it a more mea

Life systems are complex and hierarchical, with diverse components at different scales, yet they sustain themselves, grow, and evolve over time. How can a theory of such complex biological states be developed? Here we note that for a hierarchical biological system to be robust, it must achieve consistency between micro-scale (e.g. molecular) and macro-scale (e.g. cellular) phenomena. This allows for a universal theory of adaptive change in cells based on biological robustness and consistency between cellular growth and molecular replication. Here, we show how adaptive changes in high-dimensional phenotypes (biological states) are constrained to low-dimensional space, leading to the derivation of a macroscopic law for cellular states. The theory is then extended to evolution, leading to proportionality between evolutionary and environmental responses, as well as proportionality between phenotypic variances due to noise and due to genetic changes. The universality of the results across several models and experiments is demonstrated. Then, by further extending the theory of evolutionary dimensional reduction to multicellular systems, the relationship between multicellular development

Primates exhibit a robust deviation from canonical allometric scaling: at fixed body mass, their lifespans exceed those of non-primate mammals by factors of two to three. A rhesus macaque (8 kg) lives 25-40 years, whereas a cat of similar mass rarely exceeds 18 years. This statistically significant clade-level excess cannot be explained by standard metabolic or ecological models. We provide a thermodynamic explanation within the Principle of Biological Time Equivalence (PBTE), where lifespan is determined by a finite cycle budget governed by entropy production. We show that primates reduce entropy production per physiological cycle through increased neural energy allocation. The neural power fraction acts as a control parameter, extending the effective lifetime cycle count. Three mechanisms, predictive regulation, enhanced repair, and behavioral buffering, jointly suppress dissipation. This yields a quantitative neuro-metabolic multiplier that explains primate longevity and provides testable predictions linking brain energetics, entropy production, and lifespan.

Biological sequences do not come at random. Instead, they appear with particular frequencies that reflect properties of the associated system or phenomenon. Knowing how biological sequences are distributed in sequence space is thus a natural first step toward understanding the underlying mechanisms. Here we propose a new method for inferring the probability distribution from which a sample of biological sequences were drawn for the case where the sequences are composed of elements that admit a natural ordering. Our method is based on Bayesian field theory, a physics-based machine learning approach, and can be regarded as a nonparametric extension of the traditional maximum entropy estimate. As an example, we use it to analyze the aneuploidy data pertaining to gliomas from The Cancer Genome Atlas project. In addition, we demonstrate two follow-up analyses that can be performed with the resulting probability distribution. One of them is to investigate the associations among the sequence sites. This provides us a way to infer the governing biological grammar. The other is to study the global geometry of the probability landscape, which allows us to look at the problem from an evolutio

Are biological self-organising systems more ``intelligent'' than artificial intelligence (AI)? If so, why? I address this question using a mathematical framework that defines intelligence in terms of adaptability. Systems are modelled as stacks of abstraction layers (\emph{Stack Theory}) and compared by how effectively they delegate agentic control down their stacks. I illustrate this using computational, biological, military, governmental, and economic systems. Contemporary AI typically relies on static, human-engineered stacks whose lower layers are fixed during deployment. Put provocatively, such systems resemble inflexible bureaucracies that adapt only top-down. Biological systems are more intelligent because they delegate adaptation. Formally, I prove a theorem (\emph{The Law of the Stack}) showing that adaptability at higher layers is bottlenecked by adaptability at lower layers. I further show that, under standard viability assumptions, maximising adaptability is equivalent to minimising variational free energy, implying that delegation is necessary for free-energy minimisation. Generalising bioelectric accounts of cancer as isolation from collective informational structures

The stochastic exploration of the configuration space and the exploitation of functional states underlie many biological processes. The evolutionary dynamics stands out as a remarkable example. Here, we introduce a novel formalism that mimics evolution and encodes a general exploration-exploitation dynamics for biological networks. We apply it to the brain wiring problem, focusing on the maturation of that of the nematode C. elegans. We demonstrate that a parsimonious maxent description of the adult brain combined with our framework is able to track down the entire developmental trajectory.

Biological systems possess negative entropy. In them, one form of order produces another, more organized form of order. We propose a formal scheme to calculate robustness of an entire biological system by quantifying the negative entropy present in it. Our Methodology is based upon a computational implementation of two-person non-cooperative finite zero-sum game between positive (physico-chemical) and negative (biological) entropy, present in the system(TCA cycle, for this work). Biochemical analogue of Nash equilibrium, proposed here, could measure the robustness in TCA cycle in exact numeric terms, whereas the mixed strategy game between these entropies could quantitate the progression of stages of biological adaptation. Synchronization profile amongst macromolecular concentrations (even under environmental perturbations) is found to account for negative entropy and biological robustness. Emergence of synchronization profile was investigated with dynamically varying metabolite concentrations. Obtained results were verified with that from the deterministic simulation methods. Categorical plans to apply this algorithm in Cancer studies and anti-viral therapies are proposed alongsid

Exciton transfer along a polymer is essential for many biological processes, for instance light harvesting in photosynthetic biosystems. Here we apply a new witness of non-classicality to this phenomenon, to conclude that, if an exciton can mediate the coherent quantum evolution of a photon, then the exciton is non-classical. We then propose a general qubit model for the quantum transfer of an exciton along a polymer chain, also discussing the effects of environmental decoherence. The generality of our results makes them ideal candidates to design new tests of quantum features in complex bio-molecules.

We review the trade-offs between speed, fluctuations, and thermodynamic cost involved with biological processes in nonequilibrium states, and discuss how optimal these processes are in light of the universal bound set by the thermodynamic uncertainty relation (TUR). The values of the uncertainty product $\mathcal{Q}$ of TUR, which can be used as a measure of the precision of enzymatic processes realized for a given thermodynamic cost, are suboptimal when the substrate concentration $[S]$ is at the Michaelis constant ($K_\text{M}$), and some of the key biological processes are found to work around this condition. We illustrate the utility of $\mathcal{Q}$ in assessing how close the molecular motors and biomass producing machineries are to the TUR bound, and for the cases of biomass production (or biological copying processes) we discuss how their optimality quantified in terms of $\mathcal{Q}$ is balanced with the error rate in the information transfer process. We also touch upon the trade-offs in other error-minimizing processes in biology, such as gene regulation and chaperone-assisted protein folding. A spectrum of $\mathcal{Q}$ recapitulating the biological processes surveyed he

Filamentous cable bacteria exhibit unprecedented long-range biological electron transport, which takes place in a parallel fibre structure that shows an extraordinary electrical conductivity for a biological material. Still, the underlying electron transport mechanism remains undisclosed. Here we determine the intrinsic electrical properties of individual cable bacterium filaments. We retrieve an equivalent electrical circuit model, characterising cable bacteria as resistive biological wires. Temperature dependent experiments reveal that the charge transport is thermally activated, and can be described with an Arrhenius-type relation over a broad temperature range (-196°C to +50°C), thus excluding metal-like electron transport. Furthermore, when cable bacterium filaments are utilized as the channel in a field-effect transistor, they show n-type transport, indicating that electrons rather than holes are the charge carriers. Electron mobilities are in the order of 10$^{-1}$ cm$^2$/Vs, comparable to many organic semiconductors. This new type of biological centimetre-range semiconductor with low resistivity offers new perspectives for both fundamental studies and applications in (bio)e

Graph Theoretic Process Network Synthesis is described as an introduction to biological networks. Genetic, protein and metabolic systems are considered. The theoretical work of Kauffman is discussed and amplified by critical property excursions. The scaling apparent in biological systems is shown. Applications to evolution and reverse engineering are construed. The use of several programs, such as the Synprops, Design of molecules, Therm and Knapsack are suggested as instruments to study biological process network synthesis. The properties of robust self-assembly and Self-Organizing synthesis are important contributors to the discussion. The bar code of life and intelligent design is reviewed. The need for better data in biological systems is emphasized.

Biological systems are influenced by fluid mechanics at nearly all spatiotemporal scales. This broad relevance of fluid mechanics to biology has been increasingly appreciated by engineers and biologists alike, leading to continued expansion of research in the field of biological fluid dynamics. While this growth is exciting, it can present a barrier to researchers seeking a concise introduction to key challenges and opportunities for progress in the field. Rather than attempt a comprehensive review of the literature, this article highlights a limited selection of classic and recent work. In addition to motivating the study of biological fluid dynamics in general, the goal is to identify both longstanding and emerging conceptual questions that can guide future research. Answers to these fluid mechanics questions can lead to breakthroughs in our ability to predict, diagnose, and correct biological dysfunction, while also inspiring a host of new engineering technologies.

Provided that there is no theoretical frame for complex engineered systems (CES) as yet, this paper claims that bio-inspired engineering can help provide such a frame. Within CES bio-inspired systems play a key role. The disclosure from bio-inspired systems and biological computation has not been sufficiently worked out, however. Biological computation is to be taken as the processing of information by living systems that is carried out in polynomial time, i.e., efficiently; such processing however is grasped by current science and research as an intractable problem (for instance, the protein folding problem). A remark is needed here: P versus NP problems should be well defined and delimited but biological computation problems are not. The shift from conventional engineering to bio-inspired engineering needs bring the subject (or problem) of computability to a new level. Within the frame of computation, so far, the prevailing paradigm is still the Turing-Church thesis. In other words, conventional engineering is still ruled by the Church-Turing thesis (CTt). However, CES is ruled by CTt, too. Contrarily to the above, we shall argue here that biological computation demands a more ca

Biological systems, from a cell to the human brain, are inherently complex. A powerful representation of such systems, described by an intricate web of relationships across multiple scales, is provided by complex networks. Recently, several studies are highlighting how simple networks -- obtained by aggregating or neglecting temporal or categorical description of biological data -- are not able to account for the richness of information characterizing biological systems. More complex models, namely multilayer networks, are needed to account for interdependencies, often varying across time, of biological interacting units within a cell, a tissue or parts of an organism.

Many biological functions are executed by molecular machines, which consume energy and convert it into mechanical work. Biological machines have evolved to transport cargo, facilitate folding of proteins and RNA, remodel chromatin and replicate DNA. A common aspect of these machines is that their functions are driven out of equilibrium. It is a challenge to provide a general framework for understanding the functions of biological machines, such as molecular motors, molecular chaperones, and helicases. Using these machines as prototypical examples, we describe a few general theoretical methods providing insights into their functions. Although the theories rely on coarse-graining of these complex systems they have proven useful in not only accounting for many in vitro experiments but also addressing questions such as how the trade-off between precision, energetic costs and optimal performances are balanced. However, many complexities associated with biological machines will require one to go beyond current theoretical methods. Simple point mutations in the enzyme could drastically alter functions, making the motors bi-directional or result in unexpected diseases or dramatically restr

Biological signaling is imagined as a combination of activation and transport. The former is triggered by local molecular interactions and the latter is the result of molecular diffusion. However, other fundamental physical principles of communication have yet to be addressed. We have recently shown, that lipid interfaces allow for the excitation and propagation of sound pulses. Here we demonstrate, that these reversible perturbations can control the activity of membrane embedded enzymes without the necessity of molecular transport. They therefore allow for the rapid communication between distant biological entities (e.g. receptor and enzyme) at the speed of sound, which is here in the order of 1 m/s within the membrane. The mechanism reported provides a new physical framework for biological signaling.