From Cell States to Cell Fates: Control of Cell State Transitions.

We examine the coordinated behavior of thousands of genes in cell fate transitions through genome expression as an integrated dynamical system using the concepts of self-organized criticality and coherent stochastic behavior. To quantify the effects of the collective behavior of genes, we adopted the flux balance approach and developed it in a new tool termed expression flux analysis (EFA). Here we describe this tool and demonstrate how its application to specific experimental genome-wide expression data provides new insights into the dynamics of the cell-fate transitions. Particularly, we show that in cell fate change, specific stochastic perturbations can spread over the entire system to guide distinct cell fate transitions through switching cyclic flux flow in the genome engine. Utilization of EFA enables us to elucidate a unified genomic mechanism for when and how cell-fate change occurs through critical transitions.

Synchronization between Attractors: Genomic Mechanism of Cell-Fate Change.

Herein, we provide a brief overview of complex systems theory approaches to investigate the genomic mechanism of cell-fate changes. Cell trajectories across the epigenetic landscape, whether in development, environmental responses, or disease progression, are controlled by extensively coordinated genome-wide gene expression changes. The elucidation of the mechanisms underlying these coherent expression changes is of fundamental importance in cell biology and for paving the road to new therapeutic approaches. In previous studies, we pointed at dynamic criticality as a plausible characteristic of genome-wide transition dynamics guiding cell fate. Whole-genome expression develops an engine-like organization (genome engine) in order to establish an autonomous dynamical system, capable of both homeostasis and transition behaviors. A critical set of genes behaves as a critical point (CP) that serves as the organizing center of cell-fate change. When the system is pushed away from homeostasis, the state change that occurs at the CP makes local perturbation spread over the genome, demonstrating self-organized critical (SOC) control of genome expression. Oscillating-Mode genes (which normally keep genome expression on pace with microenvironment fluctuations), when in the presence of an effective perturbative stimulus, drive the dynamics of synchronization, and thus guide the cell-fate transition.

A Unified Genomic Mechanism of Cell-Fate Change.

The purpose of our studies is to elucidate the nature of massive control of the whole genome expression with a particular emphasis on cell-fate change. The whole genome expression is coordinated by the emergence of a critical point (CP: a peculiar set of biphasic genes) with the genome acting as an integrated dynamical system. In response to stimuli, the genome expression self-organizes into local sub-, near-, and super-critical states, each exhibiting distinct collective behaviors with its center of mass acting as a local attractor, coexisting with the whole genome attractor (GA). The CP serves as the organizing center of cell-fate change, and its activation makes local perturbation to spread over the genome affecting GA. The activation of CP is in turn elicited by genes with elevated temporal variance (oscillating-mode genes), normally in charge to keep genome expression at pace with microenvironment fluctuations. When oscillation exceeds a given threshold, the CP synchronizes with the GA driving genome expression state transition. The expression synchronization wave invading the entire genome is fostered by the fusion-splitting dynamics of silencing pericentromere-associated heterochromatin domains and the consequent folding-unfolding transitions of transcribing euchromatin domains. The proposed mechanism is a unified step toward a time-evolutional transition theory of biological regulation.

Self-organization of whole-gene expression through coordinated chromatin structural transition.

The human DNA molecule is a 2-m-long polymer collapsed into the micrometer space of the cell nucleus. This simple consideration rules out any "Maxwell demon"-like explanation of regulation in which a single regulatory molecule (e.g., a transcription factor) finds autonomously its way to the particular target gene whose expression must be repressed or enhanced. A gene-by-gene regulation is still more contrasting with the physical reality when in the presence of cell state transitions involving the contemporary expression change of thousands of genes. This state of affair asks for a statistical mechanics inspired approach where specificity arises from a selective unfolding of chromatin driving the rewiring of gene expression pattern. The arising of "expression waves" marking state transitions related to chromatin structural reorganization through self-organized critical control of whole-genome expression will be described in the present paper. We adopt as a model system the gene expression time course of a cancer cell (MCF-7) population exposed to an efficient stimulus causing a state transition in comparison with an ineffective stimulus. The obtained results will be put into the perspective of biological adaptive systems living on the edge of chaos.

Cell-Fate Determination from Embryo to Cancer Development: Genomic Mechanism Elucidated.

Elucidation of the genomic mechanism that guides the cell-fate change is one of the fundamental issues of biology. We previously demonstrated that whole genome expression is coordinated by the emergence of a critical point at both the cell-population and single-cell levels through the physical principle of self-organized criticality. In this paper, we further examine the genomic mechanism that determines the cell-fate changes from embryo to cancer development. The state of the critical point, acting as the organizing center of the cell fate, determines whether the genome resides in a super- or sub-critical state. In the super-critical state, a specific stochastic perturbation can spread over the entire system through the "genome engine", an autonomous critical-control genomic system, whereas in the sub-critical state, the perturbation remains at a local level. The cell-fate changes when the genome becomes super-critical. We provide a consistent framework to develop a time-evolutional transition theory for the biological regulation of the cell-fate change.

Self-Organization of Genome Expression from Embryo to Terminal Cell Fate: Single-Cell Statistical Mechanics of Biological Regulation.

A statistical mechanical mean-field approach to the temporal development of biological regulation provides a phenomenological, but basic description of the dynamical behavior of genome expression in terms of autonomous self-organization with a critical transition (Self-Organized Criticality: SOC). This approach reveals the basis of self-regulation/organization of genome expression, where the extreme complexity of living matter precludes any strict mechanistic approach. The self-organization in SOC involves two critical behaviors: scaling-divergent behavior (genome avalanche) and sandpile-type critical behavior. Genome avalanche patterns-competition between order (scaling) and disorder (divergence) reflect the opposite sequence of events characterizing the self-organization process in embryo development and helper T17 terminal cell differentiation, respectively. On the other hand, the temporal development of sandpile-type criticality (the degree of SOC control) in mouse embryo suggests the existence of an SOC control landscape with a critical transition state (i.e., the erasure of zygote-state criticality). This indicates that a phase transition of the mouse genome before and after reprogramming (immediately after the late 2-cell state) occurs through a dynamical change in a control parameter. This result provides a quantitative open-thermodynamic appreciation of the still largely qualitative notion of the epigenetic landscape. Our results suggest: (i) the existence of coherent waves of condensation/de-condensation in chromatin, which are transmitted across regions of different gene-expression levels along the genome; and (ii) essentially the same critical dynamics we observed for cell-differentiation processes exist in overall RNA expression during embryo development, which is particularly relevant because it gives further proof of SOC control of overall expression as a universal feature.

Self-Organizing Global Gene Expression Regulated through Criticality: Mechanism of the Cell-Fate Change.

BACKGROUND: A fundamental issue in bioscience is to understand the mechanism that underlies the dynamic control of genome-wide expression through the complex temporal-spatial self-organization of the genome to regulate the change in cell fate. We address this issue by elucidating a physically motivated mechanism of self-organization. PRINCIPAL FINDINGS: Building upon transcriptome experimental data for seven distinct cell fates, including early embryonic development, we demonstrate that self-organized criticality (SOC) plays an essential role in the dynamic control of global gene expression regulation at both the population and single-cell levels. The novel findings are as follows: i) Mechanism of cell-fate changes: A sandpile-type critical transition self-organizes overall expression into a few transcription response domains (critical states). A cell-fate change occurs by means of a dissipative pulse-like global perturbation in self-organization through the erasure of initial-state critical behaviors (criticality). Most notably, the reprogramming of early embryo cells destroys the zygote SOC control to initiate self-organization in the new embryonal genome, which passes through a stochastic overall expression pattern. ii) Mechanism of perturbation of SOC controls: Global perturbations in self-organization involve the temporal regulation of critical states. Quantitative evaluation of this perturbation in terminal cell fates reveals that dynamic interactions between critical states determine the critical-state coherent regulation. The occurrence of a temporal change in criticality perturbs this between-states interaction, which directly affects the entire genomic system. Surprisingly, a sub-critical state, corresponding to an ensemble of genes that shows only marginal changes in expression and consequently are considered to be devoid of any interest, plays an essential role in generating a global perturbation in self-organization directed toward the cell-fate change. CONCLUSION AND SIGNIFICANCE: 'Whole-genome' regulation of gene expression through self-regulatory SOC control complements gene-by-gene fine tuning and represents a still largely unexplored non-equilibrium statistical mechanism that is responsible for the massive reprogramming of genome expression.

Emergent Self-Organized Criticality in Gene Expression Dynamics: Temporal Development of Global Phase Transition Revealed in a Cancer Cell Line.

BACKGROUND: The underlying mechanism of dynamic control of the genome-wide expression is a fundamental issue in bioscience. We addressed it in terms of phase transition by a systemic approach based on both density analysis and characteristics of temporal fluctuation for the time-course mRNA expression in differentiating MCF-7 breast cancer cells. METHODOLOGY: In a recent work, we suggested criticality as an essential aspect of dynamic control of genome-wide gene expression. Criticality was evident by a unimodal-bimodal transition through flattened unimodal expression profile. The flatness on the transition suggests the existence of a critical transition at which up- and down-regulated expression is balanced. Mean field (averaging) behavior of mRNAs based on the temporal expression changes reveals a sandpile type of transition in the flattened profile. Furthermore, around the transition, a self-similar unimodal-bimodal transition of the whole expression occurs in the density profile of an ensemble of mRNA expression. These singular and scaling behaviors identify the transition as the expression phase transition driven by self-organized criticality (SOC). PRINCIPAL FINDINGS: Emergent properties of SOC through a mean field approach are revealed: i) SOC, as a form of genomic phase transition, consolidates distinct critical states of expression, ii) Coupling of coherent stochastic oscillations between critical states on different time-scales gives rise to SOC, and iii) Specific gene clusters (barcode genes) ranging in size from kbp to Mbp reveal similar SOC to genome-wide mRNA expression and ON-OFF synchronization to critical states. This suggests that the cooperative gene regulation of topological genome sub-units is mediated by the coherent phase transitions of megadomain-scaled conformations between compact and swollen chromatin states. CONCLUSION AND SIGNIFICANCE: In summary, our study provides not only a systemic method to demonstrate SOC in whole-genome expression, but also introduces novel, physically grounded concepts for a breakthrough in the study of biological regulation.

Global genetic response in a cancer cell: self-organized coherent expression dynamics.

Understanding the basic mechanism of the spatio-temporal self-control of genome-wide gene expression engaged with the complex epigenetic molecular assembly is one of major challenges in current biological science. In this study, the genome-wide dynamical profile of gene expression was analyzed for MCF-7 breast cancer cells induced by two distinct ErbB receptor ligands: epidermal growth factor (EGF) and heregulin (HRG), which drive cell proliferation and differentiation, respectively. We focused our attention to elucidate how global genetic responses emerge and to decipher what is an underlying principle for dynamic self-control of genome-wide gene expression. The whole mRNA expression was classified into about a hundred groups according to the root mean square fluctuation (rmsf). These expression groups showed characteristic time-dependent correlations, indicating the existence of collective behaviors on the ensemble of genes with respect to mRNA expression and also to temporal changes in expression. All-or-none responses were observed for HRG and EGF (biphasic statistics) at around 10-20 min. The emergence of time-dependent collective behaviors of expression occurred through bifurcation of a coherent expression state (CES). In the ensemble of mRNA expression, the self-organized CESs reveals distinct characteristic expression domains for biphasic statistics, which exhibits notably the presence of criticality in the expression profile as a route for genomic transition. In time-dependent changes in the expression domains, the dynamics of CES reveals that the temporal development of the characteristic domains is characterized as autonomous bistable switch, which exhibits dynamic criticality (the temporal development of criticality) in the genome-wide coherent expression dynamics. It is expected that elucidation of the biophysical origin for such critical behavior sheds light on the underlying mechanism of the control of whole genome.

Collective dynamics of specific gene ensembles crucial for neutrophil differentiation: the existence of genome vehicles revealed.

Cell fate decision remarkably generates specific cell differentiation path among the multiple possibilities that can arise through the complex interplay of high-dimensional genome activities. The coordinated action of thousands of genes to switch cell fate decision has indicated the existence of stable attractors guiding the process. However, origins of the intracellular mechanisms that create "cellular attractor" still remain unknown. Here, we examined the collective behavior of genome-wide expressions for neutrophil differentiation through two different stimuli, dimethyl sulfoxide (DMSO) and all-trans-retinoic acid (atRA). To overcome the difficulties of dealing with single gene expression noises, we grouped genes into ensembles and analyzed their expression dynamics in correlation space defined by Pearson correlation and mutual information. The standard deviation of correlation distributions of gene ensembles reduces when the ensemble size is increased following the inverse square root law, for both ensembles chosen randomly from whole genome and ranked according to expression variances across time. Choosing the ensemble size of 200 genes, we show the two probability distributions of correlations of randomly selected genes for atRA and DMSO responses overlapped after 48 hours, defining the neutrophil attractor. Next, tracking the ranked ensembles' trajectories, we noticed that only certain, not all, fall into the attractor in a fractal-like manner. The removal of these genome elements from the whole genomes, for both atRA and DMSO responses, destroys the attractor providing evidence for the existence of specific genome elements (named "genome vehicle") responsible for the neutrophil attractor. Notably, within the genome vehicles, genes with low or moderate expression changes, which are often considered noisy and insignificant, are essential components for the creation of the neutrophil attractor. Further investigations along with our findings might provide a comprehensive mechanistic view of cell fate decision.

Predicting novel features of toll-like receptor 3 signaling in macrophages.

The Toll-like receptor (TLR) 3 plays a critical role in mammalian innate immune response against viral attacks by recognizing double-stranded RNA (dsRNA) or its synthetic analog polyinosinic-polycytidylic acid (poly (IratioC)). This leads to the activation of MAP kinases and NF-kappaB which results in the induction of type I interferons and proinflammatory cytokines to combat the viral infection. To understand the complex interplay of the various intracellular signaling molecules in the regulation of NF-kappaB and MAP kinases, we developed a computational TLR3 model based upon perturbation-response approach. We curated literature and databases to determine the TLR3 signaling topology specifically for murine macrophages. For initial model creation, we used wildtype temporal activation profiles of MAP kinases and NF-kappaB and, for model testing, used TRAF6 KO and TRADD KO data. From dynamic simulations we predict i) the existence of missing intermediary steps between extracellular poly (IratioC) stimulation and intracellular TLR3 binding, and ii) the presence of a novel pathway which is essential for JNK and p38, but not NF-kappaB, activation. Our work shows activation dynamics of signaling molecules can be used in conjunction with perturbation-response models to decipher novel signaling features of complicated immune pathways.

Emergent genome-wide control in wildtype and genetically mutated lipopolysaccarides-stimulated macrophages.

Large-scale gene expression studies have mainly focused on highly expressed and 'discriminatory' genes to decipher key regulatory processes. Biological responses are consequence of the concerted action of gene regulatory network, thus, limiting our attention to genes having the most significant variations is insufficient for a thorough understanding of emergent whole genome response. Here we comprehensively analyzed the temporal oligonucleotide microarray data of lipopolysaccharide (LPS) stimulated macrophages in 4 genotypes; wildtype, Myeloid Differentiation factor 88 (MyD88) knockout (KO), TIR-domain-containing adapter-inducing interferon-beta (TRIF) KO and MyD88/TRIF double KO (DKO). Pearson correlations computed on the whole genome expression between different genotypes are extremely high (>0.98), indicating a strong co-regulation of the entire expression network. Further correlation analyses reveal genome-wide response is biphasic, i) acute-stochastic mode consisting of small number of sharply induced immune-related genes and ii) collective mode consisting of majority of weakly induced genes of diverse cellular processes which collectively adjust their expression level. Notably, temporal correlations of a small number of randomly selected genes from collective mode show scalability. Furthermore, in collective mode, the transition from large scatter in expression distributions for single ORFs to smooth linear lines emerges as an organizing principle when grouping of 50 ORFs and above. With this emergent behavior, the role of MyD88, TRIF and novel MyD88, TRIF-independent processes for gene induction can be linearly superposed to decipher quantitative whole genome differential control of transcriptional and mRNA decay machineries. Our work demonstrates genome-wide co-regulated responses subsequent to specific innate immune stimulus which have been largely neglected.

Can complex cellular processes be governed by simple linear rules?

Complex living systems have shown remarkably well-orchestrated, self-organized, robust, and stable behavior under a wide range of perturbations. However, despite the recent generation of high-throughput experimental datasets, basic cellular processes such as division, differentiation, and apoptosis still remain elusive. One of the key reasons is the lack of understanding of the governing principles of complex living systems. Here, we have reviewed the success of perturbation-response approaches, where without the requirement of detailed in vivo physiological parameters, the analysis of temporal concentration or activation response unravels biological network features such as causal relationships of reactant species, regulatory motifs, etc. Our review shows that simple linear rules govern the response behavior of biological networks in an ensemble of cells. It is daunting to know why such simplicity could hold in a complex heterogeneous environment. Provided physical reasons can be explained for these phenomena, major advancement in the understanding of basic cellular processes could be achieved.

Signaling flux redistribution at toll-like receptor pathway junctions.

Various receptors on cell surface recognize specific extracellular molecules and trigger signal transduction altering gene expression in the nucleus. Gain or loss-of-function mutations of one molecule have shown to affect alternative signaling pathways with a poorly understood mechanism. In Toll-like receptor (TLR) 4 signaling, which branches into MyD88- and TRAM-dependent pathways upon lipopolysaccharide (LPS) stimulation, we investigated the gain or loss-of-function mutations of MyD88. We predict, using a computational model built on the perturbation-response approach and the law of mass conservation, that removal and addition of MyD88 in TLR4 activation, enhances and impairs, respectively, the alternative TRAM-dependent pathway through signaling flux redistribution (SFR) at pathway branches. To verify SFR, we treated MyD88-deficient macrophages with LPS and observed enhancement of TRAM-dependent pathway based on increased IRF3 phosphorylation and induction of Cxcl10 and Ifit2. Furthermore, increasing the amount of MyD88 in cultured cells showed decreased TRAM binding to TLR4. Investigating another TLR4 pathway junction, from TRIF to TRAF6, RIP1 and TBK1, the removal of MyD88-dependent TRAF6 increased expression of TRAM-dependent Cxcl10 and Ifit2. Thus, we demonstrate that SFR is a novel mechanism for enhanced activation of alternative pathways when molecules at pathway junctions are removed. Our data suggest that SFR may enlighten hitherto unexplainable intracellular signaling alterations in genetic diseases where gain or loss-of-function mutations are observed.

Toll-like receptor signal transduction.

Toll-like receptors (TLRs) are the archetypal pattern recognition receptors in sensing exogenous pathogens. Activation of TLRs is a first line of defense of the immune system, leading to the activation and recruitment of neutrophils and macrophages to sites of infection and enhances antimicrobial activity. The TLR signaling through different intracellular molecules, such as MAP kinases and IkappaB kinases which are conserved signaling elements for many receptors, leads to a distinct set of proinflammatory gene expressions. However, how these pathways differentially and precisely control the transcription of identical genes remains largely unknown. Our review focuses on the details of up-to-date signaling molecules including negative regulators and their role in controlling innate immune response. We also stress the importance of developing systemic approaches for the global understanding of TLR signaling so that appropriate drug therapeutic targets can be identified for regulating inflammatory diseases.

Sequential logic model deciphers dynamic transcriptional control of gene expressions.

BACKGROUND: Cellular signaling involves a sequence of events from ligand binding to membrane receptors through transcription factors activation and the induction of mRNA expression. The transcriptional-regulatory system plays a pivotal role in the control of gene expression. A novel computational approach to the study of gene regulation circuits is presented here. METHODOLOGY: Based on the concept of finite state machine, which provides a discrete view of gene regulation, a novel sequential logic model (SLM) is developed to decipher control mechanisms of dynamic transcriptional regulation of gene expressions. The SLM technique is also used to systematically analyze the dynamic function of transcriptional inputs, the dependency and cooperativity, such as synergy effect, among the binding sites with respect to when, how much and how fast the gene of interest is expressed. PRINCIPAL FINDINGS: SLM is verified by a set of well studied expression data on endo16 of Strongylocentrotus purpuratus (sea urchin) during the embryonic midgut development. A dynamic regulatory mechanism for endo16 expression controlled by three binding sites, UI, R and Otx is identified and demonstrated to be consistent with experimental findings. Furthermore, we show that during transition from specification to differentiation in wild type endo16 expression profile, SLM reveals three binary activities are not sufficient to explain the transcriptional regulation of endo16 expression and additional activities of binding sites are required. Further analyses suggest detailed mechanism of R switch activity where indirect dependency occurs in between UI activity and R switch during specification to differentiation stage. CONCLUSIONS/SIGNIFICANCE: The sequential logic formalism allows for a simplification of regulation network dynamics going from a continuous to a discrete representation of gene activation in time. In effect our SLM is non-parametric and model-independent, yet providing rich biological insight. The demonstration of the efficacy of this approach in endo16 is a promising step for further application of the proposed method.

Gene expression waves. Cell cycle independent collective dynamics in cultured cells.

The ergodic hypothesis, which assumes the independence of each cell of the ensemble from all the others, is a necessary prerequisite to attach single cell based explanations to the grand averages taken from population data. This was the prevailing view about the interpretation of cellular biology experiments that typically are performed on colonies of billions of cells. By analysing gene expression data of different cells going from yeast to mammalian cell cultures, we demonstrate that cell cultures display a sort of "ecology-in-a-plate" giving rise to a rich dynamics of gene expression that are independent from reproductive cycles, hence contradicting simple ergodic assumptions The aspecific character of the observed coordinated gene expression activity inhibits any simple mechanistic hypothesis and highlights the need to consider population effects in the interpretation of data coming from cell cultures.

Advantages of external periodic events to the evolution of biochemical oscillatory reactions.

We compare, by calculations on a simple model of glycolysis, the evolutionary development of oscillatory reaction mechanisms in the presence and absence of external periodic events, such as an oscillatory or constant influx of glucose in an open reaction system. The chosen model has autonomous oscillations for given choices of the parameters of the feedback loops responsible for the oscillations, and for a given range of the total adenylate pool concentration. We change first one, then two of the parameters, so that there are no autonomous oscillations, and then vary these parameters with a genetic algorithm method in which the parameters are represented by binary strings that evolve by selection, crossover, and mutations; the optimization goal is the attainment of a high ATP/ADP concentration ratio in the system. This goal is taken to provide evolutionary advantages and is shown to be achieved more quickly in the presence of external periodic events, rather than constant influx of glucose. The results suggest the possibility of the enhanced evolutionary development of oscillatory biological reactions at shores where waves impinge on rocks and bring nutrients periodically. Measurements have shown that animals and plants grow more rapidly in the presence of such wave action than in its absence.

Neutrality condition and response law for nonlinear reaction-diffusion equations, with application to population genetics.

We study a general class of nonlinear macroscopic evolution equations with "transport" and "reaction" terms which describe the dynamics of a species of moving individuals (atoms, molecules, quasiparticles, organisms, etc.). We consider that two types of individuals exist, "not marked" and "marked," respectively. We assume that the concentrations of both types of individuals are measurable and that they obey a neutrality condition, that is, the kinetic and transport properties of the "not marked" and "marked" individuals are identical. We suggest a response experiment, which consists in varying the fraction of "marked" individuals with the preservation of total fluxes, and show that the response of the system can be represented by a linear superposition law even though the underlying dynamics of the system is in general highly nonlinear. The linear response law is valid even for large perturbations and is not the result of a linearization procedure but rather a necessary consequence of the neutrality condition. First, we apply the response theorem to chemical kinetics, where the "marked species" is a molecule labeled with a radioactive isotope and there is no kinetic isotope effect. The susceptibility function of the response law can be related to the reaction mechanism of the process. Secondly we study the geographical distribution of the nonrecurrent, nonreversible neutral mutations of the nonrecombining portion of the Y chromosome from human populations and show that the fraction of mutants at a given point in space and time obeys a linear response law of the type introduced in this paper. The theory may be used for evaluating the geographic position and the moment in time where and when a mutation originated.

Bayesian analysis of systems with random chemical composition: renormalization-group approach to Dirichlet distributions and the statistical theory of dilution.

We investigate the statistical properties of systems with random chemical composition and try to obtain a theoretical derivation of the self-similar Dirichlet distribution, which is used empirically in molecular biology, environmental chemistry, and geochemistry. We consider a system made up of many chemical species and assume that the statistical distribution of the abundance of each chemical species in the system is the result of a succession of a variable number of random dilution events, which can be described by using the renormalization-group theory. A Bayesian approach is used for evaluating the probability density of the chemical composition of the system in terms of the probability densities of the abundances of the different chemical species. We show that for large cascades of dilution events, the probability density of the composition vector of the system is given by a self-similar probability density of the Dirichlet type. We also give an alternative formal derivation for the Dirichlet law based on the maximum entropy approach, by assuming that the average values of the chemical potentials of different species, expressed in terms of molar fractions, are constant. Although the maximum entropy approach leads formally to the Dirichlet distribution, it does not clarify the physical origin of the Dirichlet statistics and has serious limitations. The random theory of dilution provides a physical picture for the emergence of Dirichlet statistics and makes it possible to investigate its validity range. We discuss the implications of our theory in molecular biology, geochemistry, and environmental science.