Entrainment and multi-stability of the p53 oscillator in human cells
The tumor suppressor p53 responds to cellular stress and activates transcription programs critical for the regulation of cell fate. DNA damage triggers a series of oscillations in p53 levels with a robust period. Guided by the theory of synchronization and entrainment we developed a model and an experimental system to test the ability of the p53 oscillator to reset and entrain to external pulses of various periods and strengths. We found that the p53 oscillator can be locked and entrained to a wide range of entrainment modes. External periods far from p53’s natural oscillations increased the heterogeneity between individual cells while stronger inputs reduced it. Single cell measurements allowed deriving the phase response curves (PRC) and multiple Arnold tongues of p53. In addition, multistability and nonlinear behaviors were detected including mode-hopping, period-doubling, and chaos. Our work revealed critical dynamical properties of p53 and provided insights into understanding and controlling it.Alba Jimenez, Alessandra Lucchetti, Mathias S. Heltberg, Liv Moretto, Carlos Sanchez, Ashwini Jambhekar, Galit Lahav, Mogens H. Jensen
Plausible, robust biological oscillations through allelic buffering
Biological oscillators control time- and dose-dependent functions via dedicated 15 oscillatory dynamics. How they achieve robust oscillations, despite the recurrent activation of noisy biochemical processes, remains unclear. Here, we characterized long-term oscillations of p53 and its feedback regulator Mdm2 in single cells after DNA damage. While p53 oscillates regularly, Mdm2 from a single MDM2 allele exhibited random unresponsiveness to ~9% of p53 pulses. Allelic-specific imaging revealed that MDM2 alleles buffer each other to maintain p53 20 pulse amplitude. Removing this buffering cripples the robustness of p53 amplitude, elevating p21 levels and cell cycle arrest. Simulations support that allelic buffering broadens evolutionary biochemical space of p53 oscillator. Our findings show how allelic buffering ensures robust p53 oscillations, highlighting its general importance for the emergence of robust biological oscillators.Feng-Shu Hsieh, Duy P. M. Nguyen, Mathias S. Heltberg, Chia-Chou Wu1, YiChen Lee, Sheng-hong Chen, Mogens H. Jensen
Local changes in potassium ions modulate dendritic integration
During neuronal activity the extracellular concentration of potassium ions ([K+]o) increases substantially above resting levels, but it remains unclear what role these [K+]o changes play in dendritic integration of synaptic inputs. We used mathematical formulations and biophysical modeling to explore the role of activity-dependent K+ changes near dendritic segments of a visual cortex pyramidal neuron, receiving synaptic inputs tuned to stimulus orientation. We found that the fine-scale spatial arrangement of inputs dictates the magnitude of [K+]o changes around the dendrites: Dendritic segments with similarly-tuned inputs can attain substantially higher [K+]o increases than segments with diversely-tuned inputs. These [K+]o elevations in turn increase dendritic excitability, leading to more robust and prolonged dendritic spikes. Ultimately, these local effects amplify the gain of neuronal input-output transformations, causing higher orientation-tuned somatic firing rates without compromising orientation selectivity. Our results suggest that local activity-dependent [K+]o changes around dendrites may act as a “volume knob” that determines the impact of synaptic inputs on feature-tuned neuronal firing.Malthe Skytte Nordentoft, Athanasia Papoutsi, Naoya Takahashi, Mathias Spliid Heltberg, Mogens Høgh Jensen, Rune Nguyen Rasmussen
Machine-learning vs. logistic regression for preoperative prediction of medical morbidity after fast-track hip and knee arthroplasty-a comparative study
Background: Machine-learning models may improve prediction of length of stay (LOS) and morbidity after surgery. However, few studies include fast-track programs, and most rely on administrative coding with limited follow-up and information on perioperative care. This study investigates potential benefits of a machine-learning model for prediction of postoperative morbidity in fast-track total hip (THA) and knee arthroplasty (TKA). Methods: Cohort study in consecutive unselected primary THA/TKA between 2014-2017 from seven Danish centers with established fast-track protocols. Preoperative comorbidity and prescribed medication were recorded prospectively and information on length of stay and readmissions was obtained through the Danish National Patient Registry and medical records. We used a machine-learning model (Boosted Decision Trees) based on boosted decision trees with 33 preoperative variables for predicting "medical" morbidity leading to LOS > 4 days or 90-days readmissions and compared to a logistical regression model based on the same variables. We also evaluated two parsimonious models, using the ten most important variables in the full machine-learning and logistic regression models. Data collected between 2014-2016 (n:18,013) was used for model training and data from 2017 (n:3913) was used for testing. Model performances were analyzed using precision, area under receiver operating (AUROC) and precision recall curves (AUPRC), as well as the Mathews Correlation Coefficient. Variable importance was analyzed using Shapley Additive Explanations values. Results: Using a threshold of 20% "risk-patients" (n:782), precision, AUROC and AUPRC were 13.6%, 76.3% and 15.5% vs. 12.4%, 74.7% and 15.6% for the machine-learning and logistic regression model, respectively. The parsimonious machine-learning model performed better than the full logistic regression model. Of the top ten variables, eight were shared between the machine-learning and logistic regression models, but with a considerable age-related variation in importance of specific types of medication. Conclusion: A machine-learning model using preoperative characteristics and prescriptions slightly improved identification of patients in high-risk of "medical" complications after fast-track THA and TKA compared to a logistic regression model. Such algorithms could help find a manageable population of patients who may benefit most from intensified perioperative care.Christian Michelsen, Christoffer C. Jørgensen*, Mathias Heltberg, Mogens H. Jensen, Alessandra Lucchetti, Pelle B. Petersen, Troels Petersen, Henrik Kehlet, Thomas Jakobsen
Enhanced DNA repair through droplet formation and p53 oscillations
Living organisms are constantly exposed to DNA damage, and optimal repair is therefore crucial. A characteristic hallmark of the response is the formation of sub-compartments around the site of damage, known as foci. Following multiple DNA breaks, the transcription factor p53 exhibits oscillations in its nuclear concentration, but how this dynamics can affect the repair remains unknown. Here, we formulate a theory for foci formation through droplet condensation and discover how oscillations in p53, with its specific periodicity and amplitude, optimize the repair process by preventing Ostwald ripening and distributing protein material in space and time. Based on the theory predictions, we reveal experimentally that the oscillatory dynamics of p53 does enhance the repair efficiency. These results connect the dynamical signaling of p53 with the microscopic repair process and create a new paradigm for the interplay of complex dynamics and phase transitions in biology.M.S. Heltberg, A. Lucchetti, F.-S. Hsieh, D.P.M. Nguyen, S.-h.Chen and Mogens H. Jensen
Coupled oscillator cooperativity as a control mechanism in chronobiology
Control of dynamical processes is vital for maintaining correct cell regulation and cell-fate decisions. Numerous regulatory networks show oscillatory behavior; however, our knowledge of how one oscillator behaves when stimulated by two or more external oscillatory signals is still missing. We explore this problem by constructing a synthetic oscillatory system in yeast and stimulate it with two external oscillatory signals. Letting model verification and prediction operate in a tight interplay with experimental observations, we find that stimulation with two external signals expands the plateau of entrainment and reduces the fluctuations of oscillations. Furthermore, by adjusting the phase differences of external signals, one can control the amplitude of oscillations, which is understood through the signal delay of the unperturbed oscillatory network. With this we reveal a direct amplitude dependency of downstream gene transcription. Taken together, these results suggest a new path to control oscillatory systems by coupled oscillator cooperativity.M.S. Heltberg, Y. Jiang, Y. Fan, Z. Zhang, M.S. Nordentoft, W. Lin, L. Qian, Q. Ouyang, Mogens H. Jensen and P. Wei
Jag1-Notch cis-interaction determines cell fate segregation in pancreatic development
The Notch ligands Jag1 and Dll1 guide differentiation of multipotent pancreatic progenitor cells (MPCs) into unipotent pro-acinar cells (PACs) and bipotent duct/endocrine progenitors (BPs). Ligand-mediated trans-activation of Notch receptors induces oscillating expression of the transcription factor Hes1, while ligand-receptor cis-interaction indirectly represses Hes1 activation. Despite Dll1 and Jag1 both displaying cis- and trans-interactions, the two mutants have different phenotypes for reasons not fully understood. Here, we present a mathematical model that recapitulates the spatiotemporal differentiation of MPCs into PACs and BPs. The model correctly captures cell fate changes in Notch pathway knockout mice and small molecule inhibitor studies, and a requirement for oscillatory Hes1 expression to maintain the multipotent state. Crucially, the model entails cell-autonomous attenuation of Notch signaling by Jag1-mediated cis-inhibition in MPC differentiation. The model sheds light on the underlying mechanisms, suggesting that cis-interaction is crucial for exiting the multipotent state, while trans-interaction is required for adopting the bipotent fate.Xiaochan Xu, Philip Allan Seymour, Kim Sneppen, Ala Trusina, Anuska la Rosa Egeskov-Madsen, Mette Christine Jørgensen, Mogens H. Jensen, Palle Serup
Coupled oscillator cooperativity as a control mechanism in chronobiology
Control of dynamical processes is vital for maintaining correct cell regulation and cell fate decisions. Numerous regulatory networks show oscillatory behavior, however, our knowledge of how one oscillator behaves when stimulated by two or more external oscillatory signals is still missing. We explore this problem by constructing a synthetic oscillatory system in yeast cells and stimulate it with two external oscillatory signals. Letting model verification and prediction operate in a tight interplay with experimental observations, we find that stimulation with two external signals expands the plateau of entrainment and reduces the fluctuations of oscillations. Furthermore, by adjusting the phase differences of the external signals, one can control the amplitude of oscillations which is understood through the signal delay of the naturally oscillatory network. With this, we reveal a direct amplitude dependency of downstream gene transcription. As a result, we discovered a path to control oscillatory systems by coupled oscillator cooperativity.Mathias S. Heltberg, Yuanxu Jiang1, Yingying Fan1, Zhibo Zhang, Malthe S. Nordentoft, Wei Lin, Long Qian, Qi Ouyang, Mogens H. Jensen, Ping Wei
Temperature controls onset and period of NF-kB oscillations and can lead to chaotic dynamics"
The transcription factor NF-κB plays a vital role in the control of the immune system, and following stimulation with TNF-α its nuclear concentration shows oscillatory behaviour. How environmental factors, in particular temperature, can control the oscillations and thereby affect gene stimulation is still remains to be resolved question. In this work, we reveal that the period of the oscillations decreases with increasing temperature. We investigate this using a mathematical model, and by applying results from statistical physics, we introduce temperature dependency to all rates, resulting in a remarkable correspondence between model and experiments. Our model predicts how temperature affects downstream protein production and find a crossover, where high affinity genes upregulates at high temperatures. Finally, we show how or that oscillatory temperatures can entrain NF-κB oscillations and lead to chaotic dynamics presenting a simple path to chaotic conditions in cellular biology.M.L. Heltberg, M. von Borries, P.M. Bendix, L.B. Oddershede and M.H. Jensen
Biophysical Modeling of Dopaminergic Denervation Landscapes in the Striatum Reveals New Therapeutic Strategy
Parkinson’s disease (PD) results from a loss of dopaminergic neurons. What triggers the break-down of neuronal signaling, and how this might be compensated, is not understood. The age of onset, progression and symptoms vary between patients, and our understanding of the clinical variability remains incomplete. In this study, we investigate this, by characterizing the dopaminergic landscape in healthy and denervated striatum, using biophysical modeling. Based on currently proposed mechanisms, we model three distinct denervation patterns, and show how this affect the dopaminergic network. Depending on the denervation pattern, we show how local and global differences arise in the activity of striatal neurons. Finally, we use the mathematical formalism to suggest a cellular strategy for maintaining normal dopamine (DA) signaling following neuronal denervation. This strategy is characterized by dual enhancement of both the release and uptake capacity of DA in the remaining neurons. Overall, our results derive a new conceptual framework for the impaired dopaminergic signaling related to PD and offers testable predictions for future research directions.M.L. Heltberg, H.N. Awada, A. Lucchetti, M.H. Jensen, J.K. Dreyer and R.N. Rasmussen
Spatial heterogeneity affects predictions from early-curve fitting of pandemic outbreaks; A case study using population data from Denmark
The modelling of pandemics has become a critical aspect in modern society. Even though artificial intelligence can help the forecast, the implementation of ordinary differential equations which estimate the time development in the number of susceptible, (exposed), infected and recovered (SIR/SEIR) individuals is still important in order to understand the stage of the pandemic. These models are based on simplified assumptions which constitute approximations, but to what extent this are erroneous is not understood since many factors can affect the development. In this paper, we introduce an agent-based model including spatial clustering and heterogeneities in connectivity and infection strength. Based on Danish population data, we estimate how this impacts the early prediction of a pandemic and compare this to the long-term development. Our results show that early phase SEIR model predictions overestimate the peak number of infected and the equilibrium level by at least a factor of two. These results are robust to variations of parameters influencing connection distances and independent of the distribution of infection rates.M.L. Heltberg, C. Michelsen, E.S. Martiny, L. Christensen, M.H. Jensen, T. Halasa, and T.C. Petersen
A tale of two rhythms: Locked clocks and chaos in biology
The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other—for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.Mathias L. Heltberg, Sandeep Krishna, Leo P. Kadanoff,and Mogens H. Jensen
Spatiotemporal model of cellular mechanotransduction via Rho and YAP
How cells sense and respond to mechanical stimuli remains an open question. Recent advances have identified the translocation of Yes-associated protein (YAP) between nucleus and cytoplasm as a central mechanism for sensing mechanical forces and regulating mechanotransduction. We formulate a spatiotemporal model of the mechanotransduction signalling pathway that includes coupling of YAP with the cell force-generation machinery through the Rho family of GTPases. Considering the active and inactive forms of a single Rho protein (GTP/GDP-bound) and of YAP (non-phosphorylated/phosphorylated), we study the cross-talk between cell polarization due to active Rho and YAP activation through its nuclear localization. For fixed mechanical stimuli, our model predicts stationary nuclear-to-cytoplasmic YAP ratios consistent with experimental data at varying adhesive cell area. We further predict damped and even sustained oscillations in the YAP nuclear-to-cytoplasmic ratio by accounting for recently reported positive and negative YAP-Rho feedback. Extending the framework to time-varying mechanical stimuli that simulate cyclic stretching and compression, we show that the YAP nuclear-to-cytoplasmic ratio’s time dependence follows that of the cyclic mechanical stimulus. The model presents one of the first frameworks for understanding spatiotemporal YAP mechanotransduction, providing several predictions of possible YAP localization dynamics, and suggesting new directions for experimental and theoretical studies.Javor K Novev, Mathias L Heltberg, Amin Doostmohammadi and Mogens H Jensen
Bifractal nature of chromosome contact maps
Modern biological techniques such as Hi-C permit one to measure probabilities that different chromosomal regions are close in space. These probabilities can be visualized as matrices called contact maps. In this paper, we introduce a multifractal analysis of chromosomal contact maps. Our analysis reveals that Hi-C maps are bifractal, i.e., complex geometrical objects characterized by two distinct fractal dimensions. To rationalize this observation, we introduce a model that describes chromosomes as a hierarchical set of nested domains and we solve it exactly. The predicted multifractal spectrum is in excellent quantitative agreement with experimental data. Moreover, we show that our theory yields a more robust estimation of the scaling exponent of the contact probability than existing methods. By applying this method to experimental data, we detect subtle conformational changes among chromosomes during differentiation of human stem cells.S. Pigolotti, Y. Zhan G. Tiana and M.H. Jensen
On chaotic dynamics in transcription factors and the associated effects in differential gene regulation
The control of proteins by a transcription factor with periodically varying concentration exhibits intriguing dynamical behaviour. Even though it is accepted that transcription factors vary their dynamics in response to different situations, insight into how this affects downstream genes is lacking. Here, we investigate how oscillations and chaotic dynamics in the transcription factor NF-kB can affect downstream protein production. We describe how it is possible to control the effective dynamics of the transcription factor by stimulating it with an oscillating ligand. We find that chaotic dynamics modulates gene expression and upregulates certain families of low-affinity genes, even in the presence of extrinsic and intrinsic noise. Furthermore, this leads to an increase in the production of protein complexes and the efficiency of their assembly. Finally, we show how chaotic dynamics creates a heterogeneous population of cell states, and describe how this can be beneficial in multi-toxic environments.Mathias L. Heltberg, Sandeep Krishna, Mogens H. Jensen
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. 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 P. Giuliani, M. H. Jensen, P.
Ditlevsen (V. Yakhot) Parametrization of Multiple Pathways in Proteins: Fast
Folding versus Growing experimental
evidence shows that proteins follow one or a few 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
J. Borg, M. H. Jensen, K. Sneppen and G. Tiana Pathways in Two-State Protein Folding. The thermodynamics of proteins indicate that
folding/unfolding A.
Hansen, K. Sneppen, M.H. Jensen Infinitely convoluted conformal mappings for fractal
aggregates. We study the interfacial growth properties of
diffusion-limited 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 J. Borg, M.H.
Jensen, K. Sneppen
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.
Tight Transitions.
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.
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.
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.
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.
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.