The Typical Set and Entropy in Stochastic Systems with Arbitrary Phase Space Growth.

The existence of the typical set is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given its central role underlying the emergence of stable, almost deterministic statistical patterns, a question arises whether typical sets exist in much more general scenarios. We demonstrate here that the typical set can be defined and characterized from general forms of entropy for a much wider class of stochastic processes than was previously thought. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces, suggesting that typicality is a generic property of stochastic processes, regardless of their complexity. We argue that the potential emergence of robust properties in complex stochastic systems provided by the existence of typical sets has special relevance to biological systems.

Modelling the dynamics of mammalian gut homeostasis.

Homeostatic balance in the intestinal epithelium relies on a fast cellular turnover, which is coordinated by an intricate interplay between biochemical signalling, mechanical forces and organ geometry. We review recent modelling approaches that have been developed to understand different facets of this remarkable homeostatic equilibrium. Existing models offer different, albeit complementary, perspectives on the problem. First, biomechanical models aim to explain the local and global mechanical stresses driving cell renewal as well as tissue shape maintenance. Second, compartmental models provide insights into the conditions necessary to keep a constant flow of cells with well-defined ratios of cell types, and how perturbations can lead to an unbalance of relative compartment sizes. A third family of models address, at the cellular level, the nature and regulation of stem fate choices that are necessary to fuel cellular turnover. We also review how these different approaches are starting to be integrated together across scales, to provide quantitative predictions and new conceptual frameworks to think about the dynamics of cell renewal in complex tissues.

Retrograde movements determine effective stem cell numbers in the intestine.

The morphology and functionality of the epithelial lining differ along the intestinal tract, but tissue renewal at all sites is driven by stem cells at the base of crypts1-3. Whether stem cell numbers and behaviour vary at different sites is unknown. Here we show using intravital microscopy that, despite similarities in the number and distribution of proliferative cells with an Lgr5 signature in mice, small intestinal crypts contain twice as many effective stem cells as large intestinal crypts. We find that, although passively displaced by a conveyor-belt-like upward movement, small intestinal cells positioned away from the crypt base can function as long-term effective stem cells owing to Wnt-dependent retrograde cellular movement. By contrast, the near absence of retrograde movement in the large intestine restricts cell repositioning, leading to a reduction in effective stem cell number. Moreover, after suppression of the retrograde movement in the small intestine, the number of effective stem cells is reduced, and the rate of monoclonal conversion of crypts is accelerated. Together, these results show that the number of effective stem cells is determined by active retrograde movement, revealing a new channel of stem cell regulation that can be experimentally and pharmacologically manipulated.

A morphospace of functional configuration to assess configural breadth based on brain functional networks.

The quantification of human brain functional (re)configurations across varying cognitive demands remains an unresolved topic. We propose that such functional configurations may be categorized into three different types: (a) network configural breadth, (b) task-to task transitional reconfiguration, and (c) within-task reconfiguration. Such functional reconfigurations are rather subtle at the whole-brain level. Hence, we propose a mesoscopic framework focused on functional networks (FNs) or communities to quantify functional (re)configurations. To do so, we introduce a 2D network morphospace that relies on two novel mesoscopic metrics, trapping efficiency (TE) and exit entropy (EE), which capture topology and integration of information within and between a reference set of FNs. We use this framework to quantify the network configural breadth across different tasks. We show that the metrics defining this morphospace can differentiate FNs, cognitive tasks, and subjects. We also show that network configural breadth significantly predicts behavioral measures, such as episodic memory, verbal episodic memory, fluid intelligence, and general intelligence. In essence, we put forth a framework to explore the cognitive space in a comprehensive manner, for each individual separately, and at different levels of granularity. This tool that can also quantify the FN reconfigurations that result from the brain switching between mental states.

Rigidity percolation uncovers a structural basis for embryonic tissue phase transitions.

Embryo morphogenesis is impacted by dynamic changes in tissue material properties, which have been proposed to occur via processes akin to phase transitions (PTs). Here, we show that rigidity percolation provides a simple and robust theoretical framework to predict material/structural PTs of embryonic tissues from local cell connectivity. By using percolation theory, combined with directly monitoring dynamic changes in tissue rheology and cell contact mechanics, we demonstrate that the zebrafish blastoderm undergoes a genuine rigidity PT, brought about by a small reduction in adhesion-dependent cell connectivity below a critical value. We quantitatively predict and experimentally verify hallmarks of PTs, including power-law exponents and associated discontinuities of macroscopic observables. Finally, we show that this uniform PT depends on blastoderm cells undergoing meta-synchronous divisions causing random and, consequently, uniform changes in cell connectivity. Collectively, our theoretical and experimental findings reveal the structural basis of material PTs in an organismal context.

Stem cell lineage survival as a noisy competition for niche access.

Understanding to what extent stem cell potential is a cell-intrinsic property or an emergent behavior coming from global tissue dynamics and geometry is a key outstanding question of systems and stem cell biology. Here, we propose a theory of stem cell dynamics as a stochastic competition for access to a spatially localized niche, giving rise to a stochastic conveyor-belt model. Cell divisions produce a steady cellular stream which advects cells away from the niche, while random rearrangements enable cells away from the niche to be favorably repositioned. Importantly, even when assuming that all cells in a tissue are molecularly equivalent, we predict a common ("universal") functional dependence of the long-term clonal survival probability on distance from the niche, as well as the emergence of a well-defined number of functional stem cells, dependent only on the rate of random movements vs. mitosis-driven advection. We test the predictions of this theory on datasets of pubertal mammary gland tips and embryonic kidney tips, as well as homeostatic intestinal crypts. Importantly, we find good agreement for the predicted functional dependency of the competition as a function of position, and thus functional stem cell number in each organ. This argues for a key role of positional fluctuations in dictating stem cell number and dynamics, and we discuss the applicability of this theory to other settings.

Decomposing information into copying versus transformation.

In many real-world systems, information can be transmitted in two qualitatively different ways: by copying or by transformation. Copying occurs when messages are transmitted without modification, e.g. when an offspring receives an unaltered copy of a gene from its parent. Transformation occurs when messages are modified systematically during transmission, e.g. when mutational biases occur during genetic replication. Standard information-theoretic measures do not distinguish these two modes of information transfer, although they may reflect different mechanisms and have different functional consequences. Starting from a few simple axioms, we derive a decomposition of mutual information into the information transmitted by copying versus the information transmitted by transformation. We begin with a decomposition that applies when the source and destination of the channel have the same set of messages and a notion of message identity exists. We then generalize our decomposition to other kinds of channels, which can involve different source and destination sets and broader notions of similarity. In addition, we show that copy information can be interpreted as the minimal work needed by a physical copying process, which is relevant for understanding the physics of replication. We use the proposed decomposition to explore a model of amino acid substitution rates. Our results apply to any system in which the fidelity of copying, rather than simple predictability, is of critical relevance.

Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems.

Understanding the thermodynamics of the duplication process is a fundamental step towards a comprehensive physical theory of biological systems. However, the immense complexity of real cells obscures the fundamental tensions between energy gradients and entropic contributions that underlie duplication. The study of synthetic, feasible systems reproducing part of the key ingredients of living entities but overcoming major sources of biological complexity is of great relevance to deepen the comprehension of the fundamental thermodynamic processes underlying life and its prevalence. In this paper an abstract-yet realistic-synthetic system made of small synthetic protocell aggregates is studied in detail. A fundamental relation between free energy and entropic gradients is derived for a general, non-equilibrium scenario, setting the thermodynamic conditions for the occurrence and prevalence of duplication phenomena. This relation sets explicitly how the energy gradients invested in creating and maintaining structural-and eventually, functional-elements of the system must always compensate the entropic gradients, whose contributions come from changes in the translational, configurational, and macrostate entropies, as well as from dissipation due to irreversible transitions. Work/energy relations are also derived, defining lower bounds on the energy required for the duplication event to take place. A specific example including real ternary emulsions is provided in order to grasp the orders of magnitude involved in the problem. It is found that the minimal work invested over the system to trigger a duplication event is around ~ 10 - 13 J , which results, in the case of duplication of all the vesicles contained in a liter of emulsion, in an amount of energy around ~ 1 kJ . Without aiming to describe a truly biological process of duplication, this theoretical contribution seeks to explicitly define and identify the key actors that participate in it.

How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions.

Sample space reducing (SSR) processes offer a simple analytical way to understand the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to search and cascading processes. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving- and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function produces the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process.

Correction: Fitting power-laws in empirical data with estimators that work for all exponents.

[This corrects the article DOI: 10.1371/journal.pone.0170920.].

The architecture of mutualistic networks as an evolutionary spandrel.

Mutualistic networks have been shown to involve complex patterns of interactions among animal and plant species, including a widespread presence of nestedness. The nested structure of these webs seems to be positively correlated with higher diversity and resilience. Moreover, these webs exhibit marked measurable structural patterns, including broad distributions of connectivity, strongly asymmetrical interactions and hierarchical organization. Hierarchical organization is an especially interesting property, since it is positively correlated with biodiversity and network resilience, thus suggesting potential selection processes favouring the observed web organization. However, here we show that all these structural quantitative patterns-and nestedness in particular-can be properly explained by means of a very simple dynamical model of speciation and divergence with no selection-driven coevolution of traits. The agreement between observed and modelled networks suggests that the patterns displayed by real mutualistic webs might actually represent evolutionary spandrels.

Chromatic transitions in the emergence of syntax networks.

The emergence of syntax during childhood is a remarkable example of how complex correlations unfold in nonlinear ways through development. In particular, rapid transitions seem to occur as children reach the age of two, which seems to separate a two-word, tree-like network of syntactic relations among words from the scale-free graphs associated with the adult, complex grammar. Here, we explore the evolution of syntax networks through language acquisition using the chromatic number, which captures the transition and provides a natural link to standard theories on syntactic structures. The data analysis is compared to a null model of network growth dynamics which is shown to display non-trivial and sensible differences. At a more general level, we observe that the chromatic classes define independent regions of the graph, and thus, can be interpreted as the footprints of incompatibility relations, somewhat as opposed to modularity considerations.

Zipf's Law, unbounded complexity and open-ended evolution.

A major problem for evolutionary theory is understanding the so-called open-ended nature of evolutionary change, from its definition to its origins. Open-ended evolution (OEE) refers to the unbounded increase in complexity that seems to characterize evolution on multiple scales. This property seems to be a characteristic feature of biological and technological evolution and is strongly tied to the generative potential associated with combinatorics, which allows the system to grow and expand their available state spaces. Interestingly, many complex systems presumably displaying OEE, from language to proteins, share a common statistical property: the presence of Zipf's Law. Given an inventory of basic items (such as words or protein domains) required to build more complex structures (sentences or proteins) Zipf's Law tells us that most of these elements are rare whereas a few of them are extremely common. Using algorithmic information theory, in this paper we provide a fundamental definition for open-endedness, which can be understood as postulates. Its statistical counterpart, based on standard Shannon information theory, has the structure of a variational problem which is shown to lead to Zipf's Law as the expected consequence of an evolutionary process displaying OEE. We further explore the problem of information conservation through an OEE process and we conclude that statistical information (standard Shannon information) is not conserved, resulting in the paradoxical situation in which the increase of information content has the effect of erasing itself. We prove that this paradox is solved if we consider non-statistical forms of information. This last result implies that standard information theory may not be a suitable theoretical framework to explore the persistence and increase of the information content in OEE systems.

Sample space reducing cascading processes produce the full spectrum of scaling exponents.

Sample Space Reducing (SSR) processes are simple stochastic processes that offer a new route to understand scaling in path-dependent processes. Here we define a cascading process that generalises the recently defined SSR processes and is able to produce power laws with arbitrary exponents. We demonstrate analytically that the frequency distributions of states are power laws with exponents that coincide with the multiplication parameter of the cascading process. In addition, we show that imposing energy conservation in SSR cascades allows us to recover Fermi's classic result on the energy spectrum of cosmic rays, with the universal exponent -2, which is independent of the multiplication parameter of the cascade. Applications of the proposed process include fragmentation processes or directed cascading diffusion on networks, such as rumour or epidemic spreading.

Fitting power-laws in empirical data with estimators that work for all exponents.

Most standard methods based on maximum likelihood (ML) estimates of power-law exponents can only be reliably used to identify exponents smaller than minus one. The argument that power laws are otherwise not normalizable, depends on the underlying sample space the data is drawn from, and is true only for sample spaces that are unbounded from above. Power-laws obtained from bounded sample spaces (as is the case for practically all data related problems) are always free of such limitations and maximum likelihood estimates can be obtained for arbitrary powers without restrictions. Here we first derive the appropriate ML estimator for arbitrary exponents of power-law distributions on bounded discrete sample spaces. We then show that an almost identical estimator also works perfectly for continuous data. We implemented this ML estimator and discuss its performance with previous attempts. We present a general recipe of how to use these estimators and present the associated computer codes.

Three faces of entropy for complex systems: Information, thermodynamics, and the maximum entropy principle.

There are at least three distinct ways to conceptualize entropy: entropy as an extensive thermodynamic quantity of physical systems (Clausius, Boltzmann, Gibbs), entropy as a measure for information production of ergodic sources (Shannon), and entropy as a means for statistical inference on multinomial processes (Jaynes maximum entropy principle). Even though these notions represent fundamentally different concepts, the functional form of the entropy for thermodynamic systems in equilibrium, for ergodic sources in information theory, and for independent sampling processes in statistical systems, is degenerate, H(p)=-∑_{i}p_{i}logp_{i}. For many complex systems, which are typically history-dependent, nonergodic, and nonmultinomial, this is no longer the case. Here we show that for such processes, the three entropy concepts lead to different functional forms of entropy, which we will refer to as S_{EXT} for extensive entropy, S_{IT} for the source information rate in information theory, and S_{MEP} for the entropy functional that appears in the so-called maximum entropy principle, which characterizes the most likely observable distribution functions of a system. We explicitly compute these three entropy functionals for three concrete examples: for Pólya urn processes, which are simple self-reinforcing processes, for sample-space-reducing (SSR) processes, which are simple history dependent processes that are associated with power-law statistics, and finally for multinomial mixture processes.

Understanding Zipf's law of word frequencies through sample-space collapse in sentence formation.

The formation of sentences is a highly structured and history-dependent process. The probability of using a specific word in a sentence strongly depends on the 'history' of word usage earlier in that sentence. We study a simple history-dependent model of text generation assuming that the sample-space of word usage reduces along sentence formation, on average. We first show that the model explains the approximate Zipf law found in word frequencies as a direct consequence of sample-space reduction. We then empirically quantify the amount of sample-space reduction in the sentences of 10 famous English books, by analysis of corresponding word-transition tables that capture which words can follow any given word in a text. We find a highly nested structure in these transition tables and show that this 'nestedness' is tightly related to the power law exponents of the observed word frequency distributions. With the proposed model, it is possible to understand that the nestedness of a text can be the origin of the actual scaling exponent and that deviations from the exact Zipf law can be understood by variations of the degree of nestedness on a book-by-book basis. On a theoretical level, we are able to show that in the case of weak nesting, Zipf's law breaks down in a fast transition. Unlike previous attempts to understand Zipf's law in language the sample-space reducing model is not based on assumptions of multiplicative, preferential or self-organized critical mechanisms behind language formation, but simply uses the empirically quantifiable parameter 'nestedness' to understand the statistics of word frequencies.

Understanding scaling through history-dependent processes with collapsing sample space.

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space-reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power laws, p(x) ~ x(-λ), where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from α = 2 to ∞. We discuss several applications showing how SSR processes can be used to understand Zipf's law in word frequencies, and how they are related to diffusion processes in directed networks, or aging processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organized critical processes.

Detection of the elite structure in a virtual multiplex social system by means of a generalised K-core.

Elites are subgroups of individuals within a society that have the ability and means to influence, lead, govern, and shape societies. Members of elites are often well connected individuals, which enables them to impose their influence to many and to quickly gather, process, and spread information. Here we argue that elites are not only composed of highly connected individuals, but also of intermediaries connecting hubs to form a cohesive and structured elite-subgroup at the core of a social network. For this purpose we present a generalization of the K-core algorithm that allows to identify a social core that is composed of well-connected hubs together with their 'connectors'. We show the validity of the idea in the framework of a virtual world defined by a massive multiplayer online game, on which we have complete information of various social networks. Exploiting this multiplex structure, we find that the hubs of the generalised K-core identify those individuals that are high social performers in terms of a series of indicators that are available in the game. In addition, using a combined strategy which involves the generalised Kcore and the recently introduced M-core, the elites of the different 'nations' present in the game are perfectly identified as modules of the generalised K-core. Interesting sudden shifts in the composition of the elite cores are observed at deep levels. We show that elite detection with the traditional K-core is not possible in a reliable way. The proposed method might be useful in a series of more general applications, such as community detection.

Towards a mathematical theory of meaningful communication.

Meaning has been left outside most theoretical approaches to information in biology. Functional responses based on an appropriate interpretation of signals have been replaced by a probabilistic description of correlations between emitted and received symbols. This assumption leads to potential paradoxes, such as the presence of a maximum information associated to a channel that creates completely wrong interpretations of the signals. Game-theoretic models of language evolution and other studies considering embodied communicating agents show that the correct (meaningful) match resulting from agent-agent exchanges is always achieved and natural systems obviously solve the problem correctly. Inspired by the concept of duality of the communicative sign stated by the swiss linguist Ferdinand de Saussure, here we present a complete description of the minimal system necessary to measure the amount of information that is consistently decoded. Several consequences of our developments are investigated, such as the uselessness of a certain amount of information properly transmitted for communication among autonomous agents.