## Mogens Høgh Jensen's research interests

Chaotic Dynamics Mediate Brain State Transitions, Driven by Changes in Extracellular Ion Concentrations

Previous studies have suggested that changes in extracellular ion concentrations initiate the transition from an activity state that characterizes sleep in cortical neurons to states that characterize wakefulness. However, because neuronal activity and extracellular ion concentrations are interdependent, isolating their unique roles during sleep-wake transitions is not possible in vivo. Here, we extend the Averaged-Neuron model and demonstrate that, although changes in extracellular ion concentrations occur concurrently, decreasing the conductance of calcium-dependent potassium channels initiates the transition from sleep to wakefulness. We find that sleep is governed by stable, self-sustained oscillations in neuronal firing patterns, whereas the quiet awake state and active awake state are both governed by irregular oscillations and chaotic dynamics; transitions between these separable awake states are prompted by ionic changes. Although waking is indicative of a shift from stable to chaotic neuronal firing patterns, we illustrate that the properties of chaotic dynamics ensure that the transition between states is smooth and robust to noise.

Rune Rasmussen, Majken Nedergaard, Mathias Heltberg, Mogens H. Jensen

Time Correlations in Mode Hopping of Coupled Oscillators

We study the dynamics in a system of coupled oscillators when Arnold Tongues overlap. By varying the initial conditions, the deterministic system can be attracted to different limit cycles. Adding external noise, the mode hopping between different states become a dominating part of the dynamics. We simplify the system through a Poincare section, and derive a 1D model to describe the dynamics. We explain that for some parameter values of the external oscillator, the time distribution of occupancy in a state is exponential and thus memoryless. In the general case, on the other hand, it is a sum of exponential distributions characteristic of a system with time correlations.

Mathias Heltberg, Sandeep Krishna and Mogens H. Jensen

Four Simple Rules that are Sufficient to Generate the Mammalian Blastocyst

Early embryonic development presents a fascinating interplay between the genetically regulated differentiation of stem cells and their larger scale organization into a blastocyst. Decades of research outlined a versatile toolbox at the core of the robust embryo development: complex intracellular networks coupled by cell-cell communication, segregation by differential adhesion, and apoptosis - are but just a few mechanisms. However, it is unclear what is the minimal set of rules sufficient for successful blastocyst development, and to what extent can these rules compensate for each other. To capture early stages of embryo development, we implemented the experimentally reported mechanisms as a set of four developmental rules in an agent-based in silico model of physically interacting cells. In a series of in silico knock-downs, we find that model reproduces many of the reported experimental phenotypes. While perturbing cell polarity results in the most severe phenotype precluding cavitation, knocking-down the apoptosis will only affect 10-30 % of the embryos. Our model pinpoints the apoptosis of the misplaced cells as a proofreading mechanism correcting for the imperfections of the "differential adhesion". We demonstrate that the complexity of the early embryo development can be captured by a minimal set of rules and interactions inferred from experimental observations. Furthermore, we quantify the degree to which rules are essential, or whether they serve to increase robustness of the unfolding developmental program.

Silas B. Nissen, Sophie M. Morgani (Rockefeller), Joshua M. Brickman (DanStem), Mogens H. Jensen, Kim Sneppen and Ala Trusina.

Tipping-point transition from transient to persistent inflammation in pancreatic islets

Type 2 diabetes (T2D) is associated with a systemic increase in the pro-inflammatory cytokine IL-1. While transient exposure to low IL-1 concentrations improves insulin secretion and beta-cell proliferation in pancreatic islets, prolonged exposure leads to impaired insulin secretion and collective beta-cell death. IL-1 is secreted locally by islet-resident macrophages and beta-cells; however it is unknown if and how the two opposing modes may emerge at single islet level. We investigated the duality of IL-1 with a quantitative in-silico model of the IL-1 regulatory network in pancreatic islets. We find that the network can produce either transient or persistent IL-1 responses, when induced by pro-inflammatory and metabolic cues. This suggests that the duality of IL-1 may be regulated at the single islet level. We use two core feedbacks in the IL-1 regulation to explain both modes: First, a fast positive feedback in which IL-1 induces its own production through IKK/NF-kB pathway. Second, a slow negative feedback where NF-kB upregulates inhibitors acting at different levels along the IKK/NF-kB pathway. A transient response ensues when the two feedbacks are balanced. When positive feedback is dominating over the negative islets transit into the persistent inflammation mode. Consistent with several observations, where the size of islets was implicated in its inflammatory state, we find that large islets and islets with high density of IL-1 amplifying cells are more prone to transit into persistent IL-1 mode. Our results are likely not limited to IL-1 but general for the combined effect of multiple pro-inflammatory cytokines and chemokines. Generalizing complex regulations in terms of two feedbacks of opposing nature and acting on different time scales provides a number of testable predictions, which call for dynamic monitoring of pro-inflammatory cytokines at the single islet level.

Thomas Holst-Hansen, Pernille Yde, Mogens H. Jensen, Thomas Mandrup-Poulsen, and Ala Trusina

Impact of Zygosity in Bimodal Phenotype Distributions

Allele number, or zygosity, is a clear determinant of gene expression in diploid cells. But the relationship between the number of copies of a gene and its expression can be hard to anticipate, especially when the gene in question is embedded in a regulatory circuit that contains feedbacks. Here we study this question making use of the natural genetic variability of human populations, which allows us to compare the expression profiles of a receptor protein in natural killer cells between donors infected with human cytomegalovirus (HCMV) with one or two copies of the allele. Crucially, the distribution of gene expression in many of the donors is bimodal, indicative of the presence of a positive feedback somewhere in the regulatory environment of the gene. Three separate gene-circuit models differing in the location of the positive feedback with respect to the gene can all reproduce well the homozygous data. However, when the resulting fitted models are applied to the hemizygous donors, only one model (the one with the positive feedback located at the level of gene transcription) reproduces the experimentally observed gene-expression profile. In that way, our work shows that zygosity can help us relate structure and function of gene regulatory networks.

Thomas Holst-Hansen, Elena Abad, Aura Muntasell, Miguel L´opez-Botet, Mogens H. Jensen, Ala Trusina, Jordi Garcia-Ojalvo

A minimal model for coupled somite precursor cells in mice

We propose a minimal model for the coupled somite precursor models in the presomitic mesoderm (PSM) in mice. The model has an internal clock based on an Axin2 negative feedback loop proposed by earlier studies \cite{jensen_wnt_2010, auleh la_wnt3a_2003}. Neighbouring cells couple through Notch concentrations, and Glycogen synthase kinase-3beta links the internal clock and the Notch concentrations in individual cells. Modeling the coupling between cells with and without delay, we simulate $3$ recent experiments. Both with and without delay in the coupling, we reproduce the qualitative results of these experiments: phase waves travelling the simulated P SM, and synchronisation of cell populations. We also find that two cell populations with initial corresponding to cells with high and low Notch concentrations respectively, synchronise out of phase with each other. This is also in agreement with recent experimental findings. For the model without delay in the cell-to-cell coupling, we simulate an experiment in which the Wnt3a-level in a cell population is varied periodically while the cells are synchronising their oscillations. We find that the cells can be e ntrained to this external variation. If period of the oscillation in Wnt3a-level is twice that of the synchronised cell population, we find that the cell population undergoes a period doubling. %Somites, precursors to vertebrae, form in the anterior presomitic mesoderm (PSM) in the vertebrate embryo. The somites form periodically, from the somitic precursor cells that constitute the PSM. This somitogenesis has been intensively stu died in mice, zebrafish and chick in particular. For these species, it has been observed that the somitic precursor cell s act like individual oscillators, coupled to their neighbours through the expression of the Notch protein \comment{Refs }. This coupling gives rise to waves of signal expression, so-called phase waves, travelling through the PSM from the po sterior to the anterior PSM. For mice and zebrafish, a new somite is formed when this wave of signal hits the anterior P SM \--- for mice, only a single wave travels through the PSM at a time, for zebrafish several waves travel at a time. In addition to this, a period gradient has been observed in the PSM of zebrafish and mice, the shortest period being in th e posterior of the PSM. This gradient has been suggested to originate from the observed signalling gradients in the PSM \--- $Wnt3a$ being on of these gradients. In mice, however, mixing a population of cells from the PSM first results in a ll cells synchronising to the mean frequency of the PSM, after which phase waves start to travel from cells that origina lly came from the posterior PSM.

Jonas Soegaard Juul, Sandeep Krishna and Mogens H. Jensen

Chaos and stochasticity in NF-kB oscillations manifest as mode hopping between entrained states

Oscillations drive many processes in biology, including transcriptional regulation by factors such as NF-kB. Cells function in dynamic environments containing periodic and aperiodic signals, and therefore it is important to understand the role of noise in controlling oscillation dynamics under fluctuating input. In such situations heterogeneity and asynchrony in NF-kB oscillation between cells may be caused both by stochasticity, as well as by deterministic chaos. Here we explore how noise and chaos interact with oscillation in the context of periodic input and show based on experiments and modeling that both cause cells to hop between entrainment modes. We find that as the amount of noise increases or the system is driven deeper into the chaotic regime, the mode-hopping becomes more frequent but the oscillations remain surprisingly well-entrained nevertheless. Thus, by functioning in regimes of strong driving, biological oscillators may avoid the deleterious effects of being in unentrained states and yet not suffer adversely from high amounts of noise or chaos.

Savas Tay (ETH), Ryan Kellogg (Stanford), Sandeep Krishna, Mathias Heltberg, Mogens H. Jensen

The role of mRNA and protein stability in the function of coupled positive and negative feedback systems in eukaryotic cells

Oscillators and switches are important elements of regulation in biological systems. These are composed of coupling negative feedback loops, which cause oscillations when delayed, and positive feedback loops, which lead to memory formation. Here, we examine the behavior of a coupled feedback system, the Negative Autoregulated Frustrated bistability motif (NAF). This motif is a combination of two previously explored motifs, the frustrated bistability motif (FBM) and the negative auto regulation motif (NAR), which both can produce oscillations. The NAF motif was previously suggested to govern long term memory formation in animals, and was used as a synthetic oscillator in bacteria. We build a mathematical model to analyze the dynamics of the NAF motif. We show analytically that the NAF motif requires an asymmetry in the strengths of activation and repression links in order to produce oscillations. We show that the effect of time delays in eukaryotic cells, originating from mRNA export and protein import, are negligible in this system. Based on the reported protein and mRNA half-lives in eukaryotic cells, we find that even though the NAF motif possesses the ability for oscillations, it mostly promotes constant protein expression at the biologically relevant parameter regimes.

Kristian Moss Bendtsen, Mogens H. Jensen, Sandeep Krishna and Szabolcs Semsey

A monomer-trimer model supports intermittent glucagon fibril growth.

We investigate in vitro fibrillation kinetics of the hormone peptide glucagon at various concentrations using confocal microscopy and determine the glucagon fibril persistence length 60mm. At all concentrations we observe that periods of individual fibril growth are interrupted by periods of stasis. The growth probability is large at high and low concentrations and is reduced for intermediate glucagon concentrations. To explain this behavior we propose a simple model, where fibrils come in two forms, one built entirely from glucagon monomers and one entirely from glucagon trimers. The opposite building blocks act as fibril growth blockers, and this generic model reproduces experimental behavior well.

Andrej Kosmrlj, Pia Cordsen, Anders Kyrsting, Daniel E. Otzen, Lene B. Oddershede and Mogens H. Jensen

Structure of a Functional Amyloid Protein Subunit Computed Using Sequence Variation.

Functional amyloid fibers, called curli, play a critical role in adhesion and invasion of many bacteria. Unlike pathological amyloids, curli structures are formed by polypeptide sequences whose amyloid structure has been selected for during evolution. This important distinction provides us with an opportunity to obtain structural insights from an unexpected source: the covariation of amino acids in sequences of different curli proteins. We used recently developed methods to extract amino acid contacts from a multiple sequence alignment of homologues of the curli subunit protein, CsgA. Together with an efficient force field, these contacts allow us to determine structural models of CsgA. We find that CsgA forms a helical structure, where each turn corresponds to previously identified repeat sequences in CsgA. The proposed structure is validated by previously measured solid-state NMR, electron microscopy, and X-ray dif- fraction data and agrees with an earlier proposed model derived by complementary means.

Pengfei Tian, Wouter Boomsma, Yong Wang, Daniel E. Otzen, Mogens H. Jensen, and Kresten Lindorff-Larsen

Long-range ordered vorticity patterns in living tissue induced by cell division.

In healthy blood vessels with a laminar blood flow, the endothelial cell division rate is low, only sufficient to replace apoptotic cells. The division rate significantly increases during embryonic development and under halted or turbulent flow. Cells in barrier tissue are con- nected and their motility is highly correlated. Here we investigate the long-range dynamics induced by cell division in an endothelial monolayer under non-flow conditions, mimicking the conditions during vessel formation or around blood clots. Cell divisions induce long-range, well-ordered vortex patterns extending several cell diameters away from the division site, in spite of the system's low Reynolds number. Our experimental results are reproduced by a hydrodynamic continuum model simulating division as a local pressure increase corresponding to a local tension decrease. Such long-range physical communication may be crucial for embryonic development and for healing tissue, for instance around blood clots.

Ninna S. Rossen, Jens M. Tarp, Joachim Mathiesen, Mogens H. Jensen and Lene B. Oddershede

Entrainment of noise-induced and limit cycle oscillators under weak noise.

Theoretical models that describe oscillations in biological systems are often either a limit cycle oscillator, where the deterministic nonlinear dynamics gives sustained periodic oscillations, or a noise-induced oscillator, where a fixed point is linearly stable with complex eigenvalues, and addition of noise gives oscillations around the fixed point with fluctuating amplitude. We investigate how each class of models behaves under the external periodic forcing, taking the well- studied van der Pol equation as an example. We find that when the forcing is additive, the noise- induced oscillator can show only one-to-one entrainment to the external frequency, in contrast to the limit cycle oscillator which is known to entrain to any ratio. When the external forcing is multiplicative, on the other hand, the noise-induced oscillator can show entrainment to a few ratios other than one-to-one, while the limit cycle oscillator shows entrain to any ratio. The noise blurs the entrainment in general, but clear entrainment regions for limit cycles can be identified as long as the noise is not too strong.

Namiko Mitarai, Uri Alon and Mogens H. Jensen

Excitable human dynamics driven by extrinsic events in massive communities. Using empirical data from a social media site (Twitter) and on trading volumes of financial securities, we analyze the correlated human activity in massive social organizations. The activity, typi- cally excited by real-world events and measured by the occurrence rate of international brand names and trading volumes, is charac- terized by intermittent fluctuations with bursts of high activity separated by quiescent periods. These fluctuations are broadly distributed with an inverse cubic tail and have long-range temporal correlations with a 1=f power spectrum. We describe the activity by a stochastic point process and derive the distribution of activity levels from the corresponding stochastic differential equation. The distribution and the corresponding power spectrum are fully consis- tent with the empirical observations.

Joachim Mathiesena, Luiza Angheluta, Peter T. H. Ahlgren, and Mogens H. Jensen

Population Genetics in Compressible Flows.

We study competition between two biological species advected by a compressible velocity field. Individuals are treated as discrete Lagrangian particles that reproduce or die in a density-dependent fashion. In the absence of a velocity field and fitness advantage, number fluctuations lead to a coarsening dynamics typical of the stochastic Fisher equation. We investigate three examples of compressible advecting fields: a shell model of turbulence, a sinusoidal velocity field and a linear velocity sink. In all cases, advection leads to a striking drop in the fixation time, as well as a large reduction in the global carrying capacity. We find localization on convergence zones, and very rapid extinction compared to well-mixed populations. For a linear velocity sink, one finds a bimodal distribution of fixation times. The long-lived states in this case are demixed configurations with a single interface, whose location depends on the fitness advantage.

Simone Pigolotti, Roberto Benzi, David R. Nelson and Mogens H. Jensen

Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli.

Inflammatory responses in eucaryotic cells are often associated with oscillations in the nuclear-cytoplasmic translocation of the transcription factor NF-kB. In most laboratory realizations, the oscillations are triggered by a cytokine stimulus. We use a mathematical model to show that an oscillatory external stimulus can synchronize the NF-kB oscillations into states where the ratios of the internal to external frequency are close to rational numbers. We predict a response diagram of the TNF-driven NF-kB system which exhibits bands of synchronization known as Arnold tongues". We suggest that when the amplitude of the external stimulus exceeds a certain threshold, chaotic dynamics of the nuclear NF-kB concentration may occur. This behaviour seems independent of the shape of the external oscillation and the nonlinearities transducing this signal.

Sandeep Krishna and Mogens H. Jensen

Cell-to-Cell Communication in Tissues: Growth, Mutations and Robustness.

In many developing tissues neighboring cells enter different developmental pathways, resulting in a fine-grained pattern of different cell states. The most common mechanism that generates such patterns is lateral inhibition, for example through Delta-Notch coupling. In this work we simulate growth of tissues consisting of a hexagonal arrangement of cells laterally inhibiting their neighbors. Mutations are performed by switching the activity of a single node and the effect is quantitative for each node in the lattice. We find that tissue growth by cell division tends to produce disordered, patchy patterns, whereas growth by migration of silent cells to the tissue results in ordered patterns. Ordered patterns are very robust to mutations of single cells, and in contrast, mutation of a cell in a disordered tissue can produce a large and widespread perturbation of the pattern. In tissues with complex patterns consisting of ordered and disordered patches, the perturbations spread along the boundaries between patches. Our work suggests that a careful experimental characterization of the disorder in tissue patterns could directly pinpoint where and how the tissue is susceptible to large-scale damage even from single cell mutations.

Benedicte Mengel, Sandeep Krishna, Sagar Chakraborty, Simone Pigolotti, Szabolcs Semsey and Mogens H. Jensen

Population genetics in compressible flows.

Organisms often grow, migrate and compete in liquid environments, as well as on solid surfaces. However, relatively little is known about genetic competitions in the presence of advecting fluid flows, despite their potential importance to past and present evolutionary history. With aquatic species such as cyanobacteria and plankton in mind, we propose and analyze a simple model of the competition between two species advected by a compressible velocity field in one dimension. The microorganisms are treated as discrete hydrodynamic Lagrangian particles that can die or reproduce via cell division. The spatial velocity configurations that act on them include a shell model of high Reynolds number turbulence, a compressible sinusoidal velocity field and a converging linear velocity that confines the microorganisms near the origin. When the amplitude of the velocity field tends to zero, the discrete number fluctuations in our model lead to the genetic demixing typical of the stepping stone model of spatial population genetics. However, both high and low Reynolds number velocity fields lead to a striking drop in the fixation time associated with neutral competitions, as well as an overall reduction carrying capacity of the medium. The converging linear velocity field leads to a bimodal distribution of fixation times. The long-lived states in this case are a genetically demixed configuration with a single boundary, whose location depends on the selective advantage.

Simone Pigolotti, Roberto Benzi, Mogens H. Jensen and David R. Nelson

Modeling the Nf-kB mediated inflammatory response predicts cytokine waves in tissue. Propagating waves have a distinct property of reliably transmitting information through space. Disctiostylium D. propagates a wave of chemoattractant to attract cells over long distances. The chemotactic response of human neutrophils -- cells that are rapidly recruited to the site of infection -- is remarkably similar to that of Dictiostyleum. While neutrophils contain all the machinery necessary to respond to the wave of chemoattractant, it is still unclear if the waves of neutrophil chemoatractant can initiate at the site of infection, what is their chemical nature and what are the cells relaying them. We propose that cytokine waves may naturally arise and propagate through the tissue as a result of NF-$\kappa$B response. Using a heuristic mathematical model of NF-$\kappa$b-like circuits coupled in space we show that many known characteristics of NF-$\kappa$b response favor cytokine wave initiation and propagation. Our model explains a number of physiological and pathological scenarios for initiation and resolution of inflammation. We find that while the propagating wave of cytokines is generally benficial for inflammation resolution, there exist special conditions favouring re-occurence of accute inflammatory response -- one of the main causes of the chronic inflammation.

Pernille Yde, Benedicte Mengel, Mogens Hogh Jensen, Sandeep Krishna, Ala Trusina

Dkk1 - a new player in Modelling the Wnt pathway. The Wnt signalling pathway transducing the stabilization of \bcatenin{} is essential for metazoan embryo development and is misregulated in many diseases such as cancer s. In recent years models have been proposed for the Wnt signalling pathway during t he segmentation process in developing embryos. Many of these include negative feedba ck loops build around Axin2 regulation. In this article we propose a new negative fe edback model for the Wnt pathway with Dickkopf1 (Dkk1) at its core. Dkk1 is a negati ve regulator of Wnt signalling. In chicken and mouse embryos there is a gradient of Wnt in the presomitic mesoderm (PSM) decreasing from the posterior to the anterior e nd. The formation of somites and the oscillations of Wnt target genes are controlled by this gradient. Here we incorporate a Wnt gradient and show that synchronization of neighbouring cells in the PSM is important in accordance with experimental observations.

Lykke Pedersen, Sandeep Krishna and Mogens H. Jensen

Emergence and Decline of Scientific Paradigms. Scientific paradigms have a tendency to rise fast and decline slowly. This asymmetry reflects the difficulty in developing a truly original idea, compared to ease at which a concept can be eroded by numerous modifications. Here we formulate a model for emergence and spread of ideas which deals with this asymmetry by constraining the ability of agents to return to already abandoned concepts. The model leads to a fairly regular pattern of global paradigm shifts, where older paradigms are eroded and subsequently replaced by new ones. The model sets the theme for a new class of pattern formation models, where local dynamics break detailed balance in a way that prevents old states to defend themselves against new nucleating or invading states. The model allows for frozen events in terms of long-term co-existence of multiple me tastable states.

Stefan Bornholdt, Kim Sneppen and Mogens H. Jensen

The Repressor-Lattice: Shared Feed-Back, Commensurability and Frustration.

A repressilator consists of a loop with three repressively interacting genes. We construct a hexagonal lattice with repressilators on all sites as a model system for many interacting feed-back loops either inside a cell or between cells. Using symmetry arguments we argue that the repressor-lattice can be in an oscillation state with only three distinct phases. Commensurability effects in the number of nodes may destroy that. Adding an activator in the center of the lattice creates frustration which causes a dynamical response that is out of balance. Adding many activators to the lattice leads to a chaotic reponse.

M. H. Jensen

Frustrated bistability as ameans to engineer oscillations in biological systems.

Oscillations play an important physiological role in a variety of biological systems. For example, respiration and carbohydrate synthesis are coupled to the circadian clock in cyanobacteria, and ultradian oscillations with time periods of a few hours have been observed in immune response (NF-kB), apoptosis (p53), development (Hes), and growth hormone secretion. Here we discuss how any bistable system can be frustrated" to produce oscillations of a desired nature by the addition of a negative feedback loop -- we use the term frustration in analogy to frustrated spins in antiferromagnets. We show that the molecular implementation can use a wide variety of methods ranging from transcription regulation to translation regulation using small non-coding RNAs to targeted protein modification. We also introduce a simple graphical method for determining whether a particular implementation will produce oscillations.

S. Krishna, S. Semsey and M. H. Jensen

Feedback and delays in the NF-kB signaling system.

The nuclear-cytoplasmic shuttling of NFkB is characterized by damped oscillations of the nuclear concentration with a time period of around an hour. The NFkB network contains at least four feedback loops passing through different regulators of NFkB activity (IkBa, IkBb, IkBe and A20) that combine to produce the overall response. However the precise role played by each feedback loop is not known. Here we construct a simple model of the system which reproduces the experimentally observed wild-type response. We then use variants of the model to examine what each feedback loop does by deleting one or more of them. We conclude that the IkBa loop drives the initial quick response and thereby hourly oscillations. In contrast, the other loops dampen the oscillations. This suggests that oscillations may not be directly physiologically important. Rather they may be a by-product of having a quick initial response. The additional feedback loops could therefore play the role of fin- tuning the response to reduce the oscillations without sacrificing the initial quick response.

B. Mengel, M. H. Jensen and S. Krishna

Stress-specific response of the p53-Mdm2 feedback loop. The p53 signalling pathway has hundreds of inputs and outputs. It can trigger cellular senescence, cell-cycle arrest and apoptosis in response to diverse stress conditions, including DNA damage, hypoxia and nutrient deprivation. Signals from all these inputs are channeled through a single node, the transcription factor p53. Yet, the pathway is flexible enough to produce different downstream gene expression patterns in response to different stresses. We construct a mathematical model of the negative feedback loop involving p53 and its inhibitor, Mdm2, at the core of this pathway, and use it to examine the effect of different stresses that trigger p53. In response to DNA damage, hypoxia, etc., the model exhibits a wide variety of specific output behaviour - steady states with low or high levels of p53 and Mdm2, as well as spiky oscillations with low or high average p53 levels. We show that even a simple negative feedback loop is capable of exhibiting the kind of flexible stress-specific response observed in the p53 system. Further, our model provides a framework for predicting the differences in p53 response to different stresses and single nucleotide polymorphisms.

A. Hunziker, S. Krishna and M. H. Jensen

Proteasome-Fibrillation Antagonism in a Dynamic Model for Parkinson's Disease.

In Parkinson's Disease (PD) there is evidence that alpha-synuclein aggregation is coupled to dysfunctional or overburdened protein quality control systems, in particular the ubiquitin-proteasome system. Here, we develop a simple dynamical model for the on-going conflict between alpha-synuclein aggregation and the maintenance of a functional proteasome in the healthy cell, based on the premise that proteasomal activity can be titrated out by mature alpha-synuclein fibrils and their protofilament precursors. In the presence of excess proteasomes the cell easily maintains homeostasis. However, when the ratio between available proteasome and the alpha-synuclein protofilaments is reduced below a threshold level, we predict a collapse of homeostasis and onset of oscillations in the proteasome concentration. Depleted proteasome opens for accumulation of oligomers. Our analysis suggest that the onset of PD is associated to a proteasome population that becomes occupied in periodic degradation of aggregates. This behavior is found to be a general state of a proteasome/chaperone system under pressure, and suggests new interpretations of other diseases where protein aggregation could stress elements of the protein quality control system.

K. Sneppen, L. Lizana, M. H. Jensen, S. Pigolotti and D. Otzen

Symbolic dynamics of biological feedback networks.

We formulate general rules for a particular coarse-graining of the dynamics, which we term symbolic dynamics', of feedback networks with monotone interactions, such as most biological modules. We show that networks which are more complex than simple cyclic structures can, in principle, show multiple, qualitatively different symbolic dynamics. Nevertheless, we show several examples where the observed symbolic dynamics is dominated by a single pattern that is very robust to changes in parameters. Further, this pattern is consistent with the dynamics being dictated by a single effective feedback loop. Our analysis provides a method for extracting these dominant feedback loops from short experimental time series, even if they only show transient trajectories.

S. Pigolotti, S. Krishna and M. H. Jensen

Oscillations and temporal signalling in cells.

The development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration. Here we review the different mechanisms which can produce oscillations in gene expression or protein concentration, using a framework of simple mathematical models. We focus on three eukaryotic genetic regulatory networks which show ultradian" oscillations, with time period of the order of hours, and involve, respectively, proteins important for development (Hes1), apoptosis (p53) and immune response (NF-kB). We argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour.

G. Tiana, S. Krishna, S. Pigolotti, M. H. Jensen and K. Sneppen

Genetic Regulation of Fluxes: Iron Homeostasis of Escherichia coli.

Iron is an essential trace-element for most organisms. However, because high concentration of free intracellular iron is cytotoxic, cells have developed complex regulatory networks that keep free intracellular iron concentration at optimal range, allowing the incorporation of the metal into iron-using enzymes and minimizing damage to the cell. We built a mathematical model of the network that controls iron uptake and usage in the bacterium Escherichia coli to explore the dynamics of iron flow. We simulate the effect of sudden decrease or increase in the extracellular iron level on intracellular iron distribution. Based on the results of simulations we discuss the possible roles of the small RNA RyhB and the Fe-S cluster assembly systems in the optimal redistribution of iron flows. We suggest that Fe-S cluster assembly is crucial to prevent the accumulation of toxic levels of free intracellular iron when the environment suddenly becomes iron rich.

S. Semsey, A. M. C. Andersson, S. Krishna, M. H. Jensen, E. Massé, and K. Sneppen

Oscillation patterns in negative feedback loops.

Organisms are equipped with regulatory systems that display a variety of dynamical behavior ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here, we study the dynamics of a very general class of negative feedback loops. Our main result is that, when a single negative feedback loop dominates the dynamical behavior, the sequence of maxima and minima of the concentrations exhibit a pattern that uniquely identifies the interactions of the loop. This allows us to devise an algorithm to (i) test whether observed oscillating time series are consistent with a single underlying negative feedback loop, and if so, (ii) reconstruct the precise structure of the loop, i.e., the activating/repressing nature of each interaction. This method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter. Physiological processes in living cells exhibit a wide

S. Pigolotti, S. Krishna, and M. H. Jensen

Kolmogorov scaling from random force fields.

We show that the classical Kolmogorov and Richardson scaling laws in fully developed turbulence are consistent with a random Gaussian force field. Numerical simulations of a shell model for turbulence suggest that the fluctuations in the force (acceleration) field are scale independent throughout the inertial regime. We find that Lagrangian statistics of the relative velocity in a turbulent flow is determined by the typical force field, whereas the multiscaling is associated to extreme events in the force field fluctuations.

M.H. Jensen, K. Sneppen and L. Angheluta

Synchronization Model for Stock Market Asymmetry.

The waiting time needed for a stock market index to undergo a given percentage change in its value is found to have an up-down asymmetry, which, surprisingly, is not observed for the individual stocks composing that index. To explain this, we introduce a market model consisting of randomly fluctuating stocks that occasionally synchronize their short term draw-downs. These synchronous events are parameterized by a fear factor'', that reflects the occurrence of dramatic external events which affect the financial market.

R. Donangelo, M. H. Jensen, I. Simonsen and K. Sneppen

Competition between Diffusion and Fragmentation: An Important Evolutionary Process of Nature.

We investigate systems of nature where the common physical processes diffusion and fragmentation compete.We derive a rate equation for the size distribution of fragments. The equation leads to a third order differential equation which we solve exactly in terms of Bessel functions. The stationary state is a universal Bessel distribution described by one parameter, which fits perfectly experimental data from two very different systems of nature, namely, the distribution of ice-crystal sizes from the Greenland ice sheet and the length distribution of alpha helices in proteins.

J. Ferkinghoff-Borg, M.H. Jensen, J. Mathiesen, P. Olesen and K. Sneppen

Exact Periodic Solutions of Shell Models of Turbulence.

We derive exact analytical solutions of the GOY shell model of turbulence. In the absence of forcing and viscosity we obtain closed form solutions in terms of Jacobi elliptic functions. With three shells the model is integrable. In the case of many shells, we derive exact recursion relations for the amplitudes of the Jacobi functions relating the different shells and we obtain a Kolmogorov solution in the limit of infinitely many shells. For the special case of six and nine shells, these recursions relations are solved giving specific analytic solutions. Some of these solutions are stable whereas others are unstable. All our predictions are substantiated by numerical simulations of the GOY shell model. From these simulations we also identify cases where the models exhibits transitions to chaotic states lying on strange attractors or ergodic energy surfaces.

P. Olesen and M. H. Jensen

A Critical Dimension'' in a Shell Model for Turbulence.

Inspired by a recently proposed theory for a critical dimension in fully
developed turbulence, we investigate the GOY shell model within this
scenario. By changing the conserved quantities, we can continuously
vary an effective dimension'' between $d=2$ and $d=3$. We identify a
criticalpoint in between these two points where the flux of energy changes
sign. Close to the critical point
the energy spectrum exhibits a scaling part and a part of thermal
equilibrium. We identify scaling laws by approaching the critical point
and perform a rescaling argument to derive a relation between
the critical exponents. We further discuss the order of the transition
and investigate the distribution function of the energy flux.
We are in the process of rewriting the GOY equations in terms
of Polyakov's generating function technique, thus obtaining
a set of coupled ordinary differential equations, and the corresponding
Fokker-Planck equation.

P. Giuliani, M. H. Jensen, P. Ditlevsen (V. Yakhot)

Parametrization of Multiple Pathways in Proteins: Fast Folding versus
Tight Transitions.

Growing experimental evidence shows that proteins follow one or a few
distinct paths when folding. We propose in this paper a procedure to
parametrize these observed pathways, and from this parametrization construct
effective Hamiltonians for the proteins. We furthermore study the
denaturated-native transitions for a wide class of possible effective Hamiltonians
based on this scheme, and find that the sharpness (tightness) of the
transitions typically areclose to their theoretical maximum and thus in quantitative
accordance with the sharp folding transition observed for single domain
proteins. Finally we demonstrate that realistic folding times are typical for the
proposed class of Hamiltonians, and we discuss the implication of the
predicted entropy barriers on the temperature dependence of the folding times.

P.G. Dommersnes, A. Hansen, M.H. Jensen, K. Sneppen

Secondary structures from hydrogen bonds in polymer folding.

We introduce a new model for hydrogen bonds in polymer
and protein folding. By considering strings made of monomers
which have both hydrogen acceptors and hydrogen donors in specific
relative orientations we demonstrate that folding of such
polymers easily lead to secondary structures in the form of helices and sheets.
This is quantified by a structure index, and is studied
as function of relative strength between the standard
Van der Walls interaction and the hydrogen binding.

J. Borg, M. H. Jensen, K. Sneppen and G. Tiana

Pathways in Two-State Protein Folding.

The thermodynamics of proteins indicate that folding/unfolding
takes place either through stable intermediates or through a
two-state process without intermediates. The rather short
folding times of the two-state process indicate that folding is guided.
We reconcile these two seemingly contradictory observations
quantitatively in a schematic model of protein folding.
We propose a new dynamical transition temperature which is lower than
the thermodynamic one, in qualitative agreement with
in vivo measurement of protein stability using {\it E.coli}.
Finally we demonstrate that our framework is easily
generalized to encompass cold unfolding, and make predictions
that relate the sharpness of the cold and hot unfolding
transitions.

A. Hansen, K. Sneppen, M.H. Jensen

Infinitely convoluted conformal mappings for fractal aggregates.

We study the interfacial growth properties of diffusion-limited
aggregation. Particularly, we use conformal growth techniques to shed
light on the screening and structures of regions bounded by the open and
branching shape characteristic for the considered type of aggregation.
The study of the mentioned regions are important for determining the
properties of the multi-fractal spectrum - properties which, until the
application of conformal growth techniques, have been difficult to
determine. In short terms our studies have so far led to a better
understanding of the emerging structures of growing aggregates.

J. Mathiesen, M.H. Jensen, (A. Levermann, I. Procaccia)

Phase transitions in long homopolymer chains.

We study the collapse of homopolymers consisting of a very long
string monomers, about 4000. Based on a new sampling technique
in Monte Carlo simulations, devised by J. Borg, we are able
to investigate the phase transitions and the ground states
to an extremely good precision. We observe three transitions:
the first is the well known collapse (theta-point) transition.
However at low temperatures we observe two first order transitions.
We speculate that the transition at the lowest temperature is
associated with an `expelling´´ of holes (voids) in the dense
packed polymer. We describe this scenario in terms of Flory
arguments following original ideas by de Gennes.

J. Borg, M.H. Jensen, K. Sneppen

Mogens Høgh Jensenmhjensen@nbi.dk

Most recently updated May 2012