Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very positive to the polemical. The other papers (Aschwanden et al. in Space Sci. Rev., 2014 , this issue; McAteer et al. in Space Sci. Rev., 2015 , this issue; Sharma et al. in Space Sci. Rev. 2015 , in preparation) in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Bak’s own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner 2012 ). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld’s original papers.

Fluctuation in signal transduction pathways is frequently observed during mammalian development. However, its role in regulating stem cells has not been explored. Here we tracked spatiotemporal ERK MAPK dynamics in human epidermal stem cells. While stem cells and differentiated cells were distinguished by high and low stable basal ERK activity, respectively, we also found cells with pulsatile ERK activity. Transitions from Basal^hi-Pulse^lo (stem) to Basal^hi-Pulse^hi, Basal^mid-Pulse^hi, and Basal^lo-Pulse^lo (differentiated) cells occurred in expanding keratinocyte colonies and in response to differentiation stimuli. Pharmacological inhibition of ERK induced differentiation only when cells were in the Basal^mid-Pulse^hi state. Basal ERK activity and pulses were differentially regulated by DUSP10 and DUSP6, leading us to speculate that DUSP6-mediated ERK pulse down-regulation promotes initiation of differentiation, whereas DUSP10-mediated down-regulation of mean ERK activity promotes and stabilizes postcommitment differentiation. Levels of MAPK1/MAPK3 transcripts correlated with DUSP6 and DUSP10 transcripts in individual cells, suggesting that ERK activity is negatively regulated by transcriptional and posttranslational mechanisms. When cells were cultured on a topography that mimics the epidermal−dermal interface, spatial segregation of mean ERK activity and pulses was observed. In vivo imaging of mouse epidermis revealed a patterned distribution of basal cells with pulsatile ERK activity, and down-regulation was linked to the onset of differentiation. Our findings demonstrate that ERK MAPK signal fluctuations link kinase activity to stem cell dynamics. Significance Understanding how intracellular signaling cascades control cell fate is a key issue in stem cell biology. Here we show that exit from the stem cell compartment in mammalian epidermis is characterized by pulsatile ERK MAPK activity. Basal activity and pulses are differentially regulated by DUSP10 and DUSP6, two phosphatases that have been shown previously to regulate differentiation commitment in the epidermis. ERK activity is controlled both transcriptionally and posttranscriptionally. Spatial segregation of mean ERK activity and pulses is observed both in reconstituted human epidermis and in mouse epidermis. Our findings demonstrate the tight spatial and temporal regulation of ERK MAPK expression and activity in mammalian epidermis.

Complex genomes show intricate organization in three-dimensional (3D) nuclear space. Current models posit that cohesin extrudes loops to form self-interacting domains delimited by the DNA binding protein CTCF. Here, we describe and quantitatively characterize cohesin-propelled, jet-like chromatin contacts as landmarks of loop extrusion in quiescent mammalian lymphocytes. Experimental observations and polymer simulations indicate that narrow origins of loop extrusion favor jet formation. Unless constrained by CTCF, jets propagate symmetrically for 1-2 Mb, providing an estimate for the range of in vivo loop extrusion. Asymmetric CTCF binding deflects the angle of jet propagation as experimental evidence that cohesin-mediated loop extrusion can switch from bi- to unidirectional and is controlled independently in both directions. These data offer new insights into the physiological behavior of in vivo cohesin-mediated loop extrusion and further our understanding of the principles that underlie genome organization.

Run-and-tumble (RnT) motion is an example of active motility where particles move at constant speed and change direction at random times. In this work we study RnT motion with diffusion in a harmonic potential in one dimension via a path integral approach. We derive a Doi-Peliti field theory and use it to calculate the entropy production and other observables in closed form. All our results are exact.

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.

The volume of phase space may grow super-exponentially (‘explosively’) with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create new emergent states. Standard ensemble theory can break down as we demonstrate in a simple model reminiscent of complex systems where new collective states emerge. We present an axiomatically defined entropy and argue that it is extensive in the micro-canonical, equal probability, and canonical (max-entropy) ensemble for super-exponentially growing phase spaces. This entropy may be useful in determining probability measures in analogy with how statistical mechanics establishes statistical ensembles by maximising entropy.