NOISE IN LIFE 2006
Workshop program
|
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Invited talks
Cooperative dynamics of interacting motors
1 Departament d’Estructura i Constituents de la Matèria, Univ. de Barcelona, Spain
2 Institut Curie, Paris, France
3 Physics Department, Technion, Haifa, Israel
4 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA
The collective dynamics of N interacting processive molecular motors are considered theoretically when an external force is applied to the leading motor. We show, using a discrete lattice model, that the force-velocity curves strongly depend on the effective dynamic interactions between motors and may differ strongly from a simple mean field prediction. Moreover, they become essentially independent of the number of motors if N is large enough (N & 5 for conventional kinesin). The general picture is essentially unaffected by the introduction of motor attachment/detachment events. These results are relevant for the interpretation of experiments of collective motors, for instance in the problem of extraction of membrane tubes out of vesicles. To gain insight into the mechanisms which generate such effective interactions, and to explore the implications in a more general perspective of Brownian motors, we study in detail a two-state ratchet model with interacting motors. We report on some exact results and systematic simulations showing that, generically, N motors interacting with only hard-core repulsion cooperate constructively, working more efficiently than N independent motors.
Transient Differentiation at the Single Cell Level
1 Division of Biology and Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125 , USA
2 Dept. de Física i Enginyeria Nuclear, Univ. Politècnica de Catalunya, Terrassa, Spain
Certain types of cellular differentiation are probabilistic and transient. In such systems individual cells can switch to an alternative state and, after some time, switch back again. In Bacillus subtilis, competence is an example of such a transiently differentiated state associated with the capability for DNA uptake from the environment. Individual genes and proteins underlying differentiation into the competent state have been identified, but it has been unclear how these genes interact dynamically in individual cells to control both spontaneous entry into competence and return to vegetative growth. Here we show that this behaviour can be understood in terms of excitability in the underlying genetic circuit. Using quantitative fluorescence time-lapse microscopy, we directly observed the activities of multiple circuit components simultaneously in individual cells, and analysed the resulting data in terms of a mathematical model. We find that an excitable core module containing positive and negative feedback loops can explain both entry into, and exit from, the competent state. We further tested this model by analysing initiation in sister cells, and by re-engineering the gene circuit to specifically block exit. Excitable dynamics driven by noise naturally generate stochastic and transient responses, thereby providing an ideal mechanism for competence regulation.
Hahn Meitner Institute Berlin, Germany
Intracellular Ca 2+ signalling exhibits a wealth of spatial and temporal structures which are generated by noise in channel state behavior. We show the theoretical considerations predicting that mechanism, the experimental confirmation and first steps of modelling following from that concept.
Circadian rhythms and molecular noise
Faculté des Sciences, Université Libre de Bruxelles, Belgium
Circadian rhythms, characterized by a period of about 24h, are the most widespread biological rhythms generated autonomously at the molecular level. The core molecular mechanism responsible for circadian oscillations relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models account for the occurrence of autonomous circadian oscillations, for their entrainment by light-dark cycles, and for their phase shifting by light pulses. Stochastic versions of these models take into consideration the molecular fluctuations that arise when the number of molecules involved in the regulatory mechanism is low. Numerical simulations of the stochastic models show that robust circadian oscillations can already occur with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively. Various factors affect the robustness of circadian oscillations with respect to molecular noise. Besides an increase in the number of molecules, entrainment by light-dark cycles and cooperativity in repression enhance robustness, whereas the proximity of a bifurcation point leads to less robust oscillations. Another parameter that appears to be crucial for the robustness of circadian rhythms is the binding/unbinding rate of the inhibitory protein to the promoter of the clock gene. Intercellular coupling further increases the robustness of circadian oscillations.
Multistability and noise in T cell gene expression
Institute of Biology, Humboldt University Berlin, Germany
Proliferation and differentiation of immune cells are controlled by diffusible cytokine signals. In this talk I will discuss the dynamics of a gene-regulatory network that governs the expression T cell growth factor, interleukin-2 (IL-2). This cytokine stimulates the proliferation of both conventional T cells that activate immune responses and the recently discovered regulatory T cells (Treg) that inhibit immune responses. How can IL-2 reliably elicit one or the other effect? Based on T-cell culture experiments, we have developed a reaction-diffusion model describing the production of IL-2 by conventional T cells challenged with antigen and the competition for IL-2 by Treg. Positive feedbacks exterted by IL-2 on its own production and the expression of its plasma-membrane receptors cause bistability and an associated all-or-none behavior of T cell activation. We have gained experimental support for this prediction by demonstrating that, depending on the extent of feedback, the expression of IL2 receptors on a cell can follow an all-or-none or a graded pattern. Interestingly, the model shows that, despite the high diffusivity of the protein, communication through IL-2 occurs only between neighbouring cells. The parameters responsibe for this localization of signaling are identified. The mechanism of local competition for the cytokine between T cells and Treg allows a consistent interpretation of what have previously been considered conflicting experimental findings and may underly specificity of IL-2 action in immune regulation.
Finding the Center Reliably: Robust Patterns of Developmental Gene Expression
1 Department of Mathematics, Imperial College London, United Kingdom
2 AMOLF Institute Amsterdam, The Netherlands
We investigate a mechanism for the robust identification of the center of a developing biological system. We assume the existence of two morphogen gradients, an activator emanating from the anterior, and a corepressor from the posterior. The corepressor inhibits the action of the activator in switching on target genes. We apply this system to Drosophila embryos, where we predict the existence of a hitherto undetected posterior corepressor. Using mathematical modeling, we show that a symmetric activator-corepressor model can quantitatively explain the precise mid-embryo expression boundary of the hunchback gene, and the scaling of this pattern with embryo size.
Transcriptional noise and its manifestations in the somite segmentation clock
Vertebrate Development Lab, Cancer Research UK LRI, 44 Lincoln's Inn Fields, London WC2A 3PX
The control of gene transcription is a noisy process, and it can be a powerful source of randomness in cell behaviour. The essentially stochastic nature of gene regulation, with each gene copy flickering between active and inactive states as a result of an association/dissociation reaction with a gene regulatory protein, can become manifest in the gross behaviour of the cell through the enormous amplification brought about by transcription and translation. The somite segmentation clock - the transcriptional oscillator that controls the spacing of somites (body segments) in the vertebrate embryo - is one system where such stochastic effects may have important consequences.
The clock operates in the presomitic mesoderm (PSM) at the tail end of the embryo; here, the cells show synchronized oscillations in the expression of certain genes. In each oscillator cycle, one additional somite is delimited and emerges as a block of cells from the PSM. In the zebrafish, the clock has a period of 30 minutes and involves oscillating expression of her1, her7, deltaC and some other genes. Synchrony between adjacent cells is dependent on cell-cell communication via the Delta-Notch signalling pathway; mutations in this pathway disrupt synchronisation and cause gross defects in body segmentation. The mechanism of the oscillation can be plausibly explained in terms of a negative feedback loop though which Her7 (or perhaps Her1) directly inhibits its own expression; the oscillation period, according to this model, is mainly determined by the transcriptional delay - the time from start to finish of the synthesis of a her7 mRNA molecule. The model explains how Notc!
h signalling normally maintains synchrony between adjacent cells, and it relates the stochastic variability in the free-running rhythms of individual uncoupled cells to the value of koff for the dissociation of Her7 protein from its binding site in the regulatory region of the her7 gene. We are currently testing the model experimentally and measuring the parameters on which its behaviour depends.
Coupled dynamics of DNA conformations and DNA-binding proteins
NORDITA – Nordic Institute for Theoretical Physics, Copenhagen, Denmark
Despite being the thermodynamically stable state of DNA under standard salt conditions, double-stranded DNA can locally denature and expose flexible single-stranded bubbles. Based on some recent single DNA experiments, I will present a simple model for the opening and closing dynamics of these bubbles based on the well-established Poland-Scheraga model [1]. Complimentary to this master equation approach, a stochastic simulation technique based on the Gillespie algorithm will also be introduced [2]. In particular, I will show how, and under what conditions, the time scale competition between DNA bubbles and the binding of selectively single-stranded DNA binding proteins prevents complete denaturation by these proteins (kinetic block) [3]. The majority of biological functions of DNA rely on site-specific DNA-binding proteins finding their targets, and therefore searching efficiently through megabases of non-target DNA. A particular case is gene expression. I will introduce some recent advances in the understanding of the target search, and how it relates to the local and global conformations of the DNA molecule. Starting from the first measurement of a purely one-dimensional search process, where the binding protein find its target while sliding on the DNA before it detaches on average [4], I will discuss the coupling of different search mechanisms: Three-dimensional volume diffusion, the same one-dimensional search while non-specifically bound to the DNA [5], and finally a Lévy flight component due to intersegmental transfer at DNA-loops [6]. The model exhibits a rich variety of scaling behaviours.
T. Ambjörnsson and R. Metzler, “Binding dynamics of single-stranded DNA binding proteins to fluctuating bubbles in breathing DNA”, J. Phys. Cond. Mat. 17 , S1841 (2005).
S. K. Banik, T. Ambjörnsson, and R. Metzler, “Stochastic approach to DNA breathing dynamics” Europhys. Lett. 71 , 852 (2005).
R. Metzler and T. Ambjörnsson, “Coupled dynamics of DNA breathing and of proteins that selectively bind to single-stranded DNA”, Phys. Rev. E 72 , 030901(R) (2005).
I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams, “Target search of N sliding proteins on a DNA”, Biophys. J. 89 , 895 (2005).
A. Bakk and R. Metzler, “In vivo non-specific binding of λ CI and Cro repressors is significant”, FEBS Lett. 563 , 66 (2004).
M. A. Lomholt, T. Ambjörnsson, and R. Metzler, “Optimal target search on a fast folding polymer chain with volume exchange”, Phys. Rev. Lett. (2006), at press; E-print cond-mat/0510072.
Relaxation in nonlinear elastic networks and prototypes of molecular machines
Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Berlin, Germany
By running an evolutionary optimization process, we construct nonlinear elastic networks with special relaxation properties, able to operate similar to protein machines. An example of an artificial machine-like device, with the cycle powered by binding of a ligand, is presented and its stochastic behavior is discussed. This research may help to understand the design principles of molecular motors and provide directions for the construction of artificial nonequilibrium nanodevices functioning as motors and machines.
Noisy cellular decision making: from temporal to spatial choices
Department of Physics, Massachusetts Institute of Technology
Feedback regulation is often used in gene and protein networks to define a stable decision switch. I will discuss two example of networks that use this feedback strategy in both a temporal and spatial sense.
First, I will discuss the galactose signaling pathway in budding yeast. This network contains multiple nested feedback loops. From the two positive feedback loops only the Gal3p-mediated loop is able to generate two stable expression states with a persistent memory of previous galactose consumption states. The parallel, Gal2p-mediated loop only increases the expression difference between the two states. A negative feedback through Gal80p reduces the strength of the core positive feedback. Despite this fact, a constitutive increase of the Gal80p concentration tunes the system from having destabilized memory to persistent memory. A model reveals that fluctuations are trapped more efficiently at elevated Gal80p levels. Indeed, the rate at which single cells randomly switch back-and-forth between expression states, was reduced.
Second, I will discuss a protein network that utilizes feedback regulation to control spontaneous cell polarization in budding yeast. Cellular polarization is often a response to distinct extracellular or intracellular cues, such as nutrient gradients or cortical landmarks. However, in the absence of such cues, some cells can still select a polarization axis at random. Positive feedback loops promoting localized activation of the GTPase Cdc42p are central to this process. I will discuss spontaneous polarization during bud site selection in yeast cells that lack functional landmarks. We find that these cells do not select a single random polarization axis, but continuously change this axis during the G1 phase of the cell cycle. This is reflected in traveling waves of activated Cdc42p which randomly explore the cell periphery. Our integrated computational and in vivo analyses of these waves reveal a negative feedback loop that competes with the positive feedback loops to regulate Cdc42p activity and confer dynamic responsiveness on the robust initiation of cell polarization.
Fundamental limits and trade-offs in negative feedback control
Department of Systems Biology, Harvard Medical School, USA
The conventional wisdom on negative feedback in biology is that it checks spontaneous fluctuations and provides homeostasis to external changes. We challege this view by showing how different sources of variation -spontaneous low-copy fluctuations, feedback noise, and external changes- are suppressed according to incompatible principles, generating frustration trade-offs where reducing one type of variation amplifies another. The trade-offs produce physical limits to noise suppression that can be formulated in terms of the biological constraints. We further show how approaching the limits can require exotic features that have been widely overlooked in biology, including counteracting feedback loops, accelerating gains, and non-Markovian molecular memory.
Contributed abstracts
Dorsoventral boundary formation in Drosophila ’s wing imaginal disks
Parc Científic de Barcelona, Centre de Recerca en Química Teòrica (CeRQT), C/ Josep Samitier 1-5 08028 Barcelona
Gene expression underlies morphogenetic processes and provides positional information to cells. In this way, the genetic activity sets up a “map” by means of which the cell fate is univocally determined in terms of the relative position on the cells within the primordium. This pattern formation process is utterly illustrated in the Drosophila ’s imaginal disks. Within this framework, we have focused on the establishment of the dorsoventral (DV) border of the wing imaginal disk. Such border helps to organize the primordium and provides signaling between/to compartments during development. It is known that the notch signaling pathway is key to the understanding of the formation of the DV axis. However a number of open problem remain. Namely, the restriction of the border to two, or three, cells, the larger notch activity where wingless (a notch antagonist) expression is maximum, and the process that leads to a symmetric expression of notch ligands (delta and serrate) at both sides of the border starting from an asymmetric situation. Herein we present a model for a gene regulatory network that explains these open problems altogether. In particular, we show that the concept of refractoriness is required to explain properly the establishment of the DV axis. Our in silico results are complemented by in vivo experiments that support our findings.
Multistable and multistep dynamics in neutrophil differentiation
Program in Biophysics, Harvard University; Vascular Biology Program, Childrens' Hospital, Boston
Cell differentiation has long been theorized to represent a switch in a bistable system, and recent experimental work in micro-organisms has revealed bistable dynamics in small gene regulatory circuits. However, the dynamics of mammalian cell differentiation has not been analyzed with respect to bistability.
Here we studied how HL60 promyelocytic precursor cells transition to the neutrophil cell lineage after stimulation with the differentiation inducer, dimethyl sulfoxide (DMSO). Single cell analysis of the expression kinetics of the differentiation marker CD11b (Mac-1) revealed all-or-none switch-like behavior, in contrast to the seemingly graduated change of expression when measured as a population average. Progression from the precursor to the differentiated state was detected as a discrete transition between low (CD11bLow) and high (CD11bHigh) expressor subpopulations distinguishable in a bimodal distribution. Hysteresis in the dependence of CD11b expression on DMSO dose suggests that this bimodality may reflect a bistable dynamic. But when an unswitched (CD11bLow ) subpopulation of cells in the bistable/bimodal regime was isolated and cultured, these cells were found to differ from undifferentiated precursor cells in that they were primed to differentiate.
These findings indicate that differentiation of human HL60 cells into neutrophils does not result from a simple state transition of a bistable switch as traditionally modeled. Instead, mammalian differentiation appears to be a multi-step process in a high-dimensional system, a result which is consistent with the high connectivity of the cells complex underlying gene regulatory network.
Kinesin as an electrostatic machine
1 Departament d’Estructura i Constituents de la Matèria, Univ. de Barcelona, Spain
2 Department of Physics, University of Crete and Institute of Electronic Structure and Laser, Heraklion, Greece
Kinesin is a two-headed motor protein that uses ATP hydrolysis to propel itself on tubulin microtubules advancing in 8nm discrete steps. We show here through numerical calculations on the involved electro-chemical phenomena that some of the accepted experimental features, also the most recent ones, can be explained by considering that kinesin is an electrostatic motor. We find that it is not necessary to assume global conformational changes. The specific directional motion is made possible by two symmetry breaking physical processes related to the net electric polarity of tubulin and to the fact that the amino-acidic character of the stalk domain provides a specific charge to the neck. Our calculations show that the net microtubule dipole moment as well as the neck charge determines whether the motor is plus (wild-type) or minus-ended (ncd). The three-dimensional electric field induced by the tubulin dipole moment forces the motor to walk parallel to a protofilament. Additionally, we discuss the role of fluctuations on the mechanics of the stepping.
Mesoscopic methods for noisy biochemical processes
1 Centre for Mathematics and Computer Science, Dep. Modelling, Analysis and Simulation, Amsterdam, The Netherlands
2 University of Amsterdam, Section Computational Science, Amsterdam, The Netherlands
In this talk we discuss the simulation of stochastic effects in biochemical processes with the currently available mesoscopic methods. We apply them to simple models: gene expression and bi-stable biochemical systems. We show the range of validity of the methods depending on the number of biomolecules involved in a process. We look at the averages and variances (noise) of product levels obtained with these methods. There are three groups of methods used for modelling living systems. The regimes in which they are valid are overlapping and it is often not clear which model should be chosen to study the stochastic phenomena at the mesoscopic level. The lowest reasonable description, regarding CPU-power, is based on the Smoluchowski diffusion equation, which has a long tradition in modelling chemical reactions. Brownian dynamics - a numerical scheme to solve the Smoluchowski diffusion equation, is an expensive method to tackle the relevant time scales in biochemical networks. Therefore, recent methods based on this model, i.e. Green’s function reaction dynamics (GFRD) [1] and Smoluchowski dynamics (Smoldyn) [2] allow for a more effective explicit simulation of biomolecules in space and time. Another group of computational methods such as MesoRD [3] or the Gillespie Multi-Particle method [4] is based on Reaction-Diffusion Master Equation (RDME). In this framework space is discretised into volumes in which discrete substrates are assumed to be well-mixed in space. Therefore the resolution of the reactant localisation is coarser but the efficiency is significantly higher. Finally we look at the reaction-diffusion partial differential equation model. It is by far the cheapest way to model spatio-temporal phenomena in life systems but contrary to the previous ones it does not account for the stochastic effects, nor the discrete nature of chemical components.
J.S. van Zon and P.R. ten Wolde, Simulating Biochemical Networks at the Particle Level and in Time and Space: Green’s Function Reaction Dynamics, Phys. Rev. Lett. 94, 128103 (2005).
S.S. Andrews and D. Bray, Stochastic simulation of chemical reactions with spatial resolution and single molecule detail, Phys. Biol. 1, 137–151 (2004).
J. Hattne, D. Fange and J. Elf, Stochastic reaction-diffusion simulation with MesoRD, Bioinformatics, 21 (12), 2923–2924 (2005).
J.V. Rodríguez, J.A. Kaandorp, J.G. Blom, Spatial Stochastic Modelling of the phosphoenolpyruvatedependent phosphotransferase (PTS) pathway in Escherichia coli, submitted (2006).
The covalent cycle as a tunable low-pass filter that is robust to noise
1 Harvard-MIT Division of Health Science and Technology & Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Cells rely on signaling pathways, which are built of proteins subject to various sources of noise, to sense, transduce and process information from their environment. A common motif in these pathways is the covalent modification cycle that consists of a (substrate) protein that becomes active when phosphorylated by an active kinase and that is rendered inactive by a phosphatase that removes the phosphate group. From a systems perspective, one may think of the number of kinases as the input and the number of active proteins as the output of the system. Covalent cycles experience fluctuations in the number of active kinases (input noise), cell-to-cell variability in the concentration of substrate protein, (extrinsic noise), and to fluctuations due to the intrinsic stochastic nature of chemical reactions (intrinsic noise). What function do these cycles serve that makes them so pervasive in signaling pathways? And how robust are they to the different noise sources that they are subjected to? We model the covalent modification cycle and focus on a parameter regime where the kinase is saturated but the phosphatase is unsaturated, which we call the signal-transducing regime. In this regime, we find that the cycle acts as a low-pass filter that faithfully transmits temporal changes in the kinase activity below a tunable cut-off frequency, while filtering out high-frequency components in the input (the input noise). Furthermore, the activity of the cycle is largely independent of concentration of substrate protein, making the switch robust to extrinsic noise. Lastly, we show that the instrinsic noise causes a signal-tonoise ratio in the output that is proportional to the square root of the number of active proteins, giving the cycle increasing robustness to intrinsic noise with increasing kinase activity. Our results show that covalent modification cycles, when in the proper regime, are ideal signal-transducing modules: they act as tunable low-pass filters of kinase activity that are robust not only to high-frequency noise in the input, but also to extrinsic and intrinsic noise. These very desirable properties may help explain the pervasiveness of covalent modification cycles in signaling pathways, as well as the impact and role of different types of noise in cell signaling.
Genetic design and dynamics of minimal cellular oscillators
1 Instituto 'Nicolás Cabrera', Facultad de Ciencias C-XVI, Universidad Autónoma de Madrid, Spain
2 Evolutionary Systems Biology Initiative, Structural and Computational Biology Programme, Spanish National Cancer Center (CNIO), Madrid, Spain
Genetic oscillators based on the interaction of a small set of molecular components have been shown to be involved in the regulation of the cell cycle, the circadian rhythms or the response of several signaling pathways. Uncovering the functional properties of such oscillators becomes thus important for the understanding of these cellular processes, and for the characterization of fundamental properties of more complex clocks [1].
One key aspect is to establish the role played by the specific 'design' or genetic circuit architecture in the functioning of the oscillator. In this contribution, we show how the dynamics of a minimal two-component oscillator is drastically affected by its genetic implementation [2]. We consider a repressor and activator element combined in a simple logical motif. While activation is always exerted at the transcriptional level, repression is alternatively operating at the transcriptional (design I) or post-translational (design II) level. These designs display differences on basic oscillatory features and on their behavior with respect to molecular noise or entrainment by periodic signals. In particular, design I induces oscillations with large activator amplitudes and arbitrarily small frequencies, and acts as an "integrator" of external stimuli, while design II shows emergence of oscillations with finite, and less variable, frequencies and smaller amplitudes, and detects better frequency-encoded signals ("resonator"). Similar types of stimulus response are observed in neurons and thus this work enable us to connect very different biological contexts. Our findings are relevant for the characterization of the physiological roles of simple oscillator motifs, the understanding of core machineries of complex clocks, and the bio-engineering of synthetic oscillatory circuits.
J. Hasty, D. McMillen and J. J. Collins, “Engineered gene circuits”, Nature 420 , 224 (2002).
R. Guantes and J F Poyatos, “Genetic design principles of two-component cellular oscillators”, PLoS Computational Biology, submitted.
Stochastic analysis of simple chemical reactions
Department of Bioengineering, Imperial College London, United Kingdom
The specific characteristics of cellular biochemistry have made stochastic descriptions of chemical reactions increasingly necessary. However, the corresponding Master Equations (MEs) are hard to solve analytically and approximate solutions and numerical algorithms have been developed to deal with these systems. In order to evaluate the performance of different methods, we obtain analytical results for the stationary solution of the ME of simple reactions. In addition, we obtain the complete time-dependent ME solution for a simple irreversible first order reaction. We then use our analytical results to investigate the accuracy of approximate solutions (the Fokker-Planck Equation and van Kampen’s Linear Noise Approximation) and numerical methods (the Gillespie algorithm and direct numerical solutions) for the solution of the MEs. We find that the direct numerical solution is superior to the other methods in simple systems. We also quantify the convergence of Gillespie’s Monte Carlo algorithm and show that the L2 norm of the error decreases as √ N, where N is the number of simulations. By employing the Coupling From The Past algorithm, we guarantee exact sampling from the stationary distribution thus increasing the accuracy of the Gillespie algorithm. In addition to investigating the convergence of the probability function, we study related quantities such as central moments and first passage times, which are of practical interest in biological experiments.
Cooperative escape dynamics of polymer chains under microcanonical conditions
Institut für Physik, Humboldt Universität Berlin, Germany
We consider the self-organized escape of a onedimensional chain of coupled units from a metastable state over a barrier. The underlying dynamics is conservative and deterministic. For an almost uniform and linear lattice state where all the units are initially situated near the bottom the dynamics seems to be restrained preventing escape from the well. It is demonstrated that internal energy redistribution leads to such strong localization that a critical localized mode, viz. a transition state is formed. The latter, being dynamically unstable, constitutes the starting point for the collective escape process which proceeds as kink-antikink motions along the chain. We consider applications of the collective escape feature to fragmentation and translocation processes of macromolecules.
Cell lineage transport: a mechanism for molecular gradient formation
1 Department d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Spain
2 Gene Expression Laboratory, The Salk Institute for Biological Studies, La Jolla, 92037 CA, USA
Gradient formation by secreted proteins (morphogens) that move along an existing field of cells is a fundamental patterning mechanism during embryo development [1-3]. The most widely recognized mechanism for protein gradient formation is the secretion of a protein that passively diffuses away from its source of production, although other molecular transport mechanisms can be participating [4-6]. However, transport mechanisms capable of generating gradients and involving cell dynamics, cell proliferation, cell growth, or cell migration, have not been analyzed in detail, especially from a theoretical point of view [7-9]. Importantly, these transport mechanisms can elicit the formation of gradients of mRNA and non-secreted proteins, and can be extended to secreted molecules. Here we address mathematically the feasibility and the properties of gradients of mRNA and proteins formed by molecular dilution and cell transport in growing tissues, a mechanism we termed “cell lineage transport”. The emergence of these gradients involves several stochastic processes: bursts of mRNA transcription, fluctuations in the distribution of the molecular content among the two daughters of a dividing cell, and non-constant periods of the cell cycle [10]. Our mathematical analysis reveals that these gradients exhibit a power-law profile, arising from the non-linear cellular transport. In addition, these gradients are rather robust to fluctuations in the duration of the cell cycle and in the molecular dilution, and exhibit size-scaling properties. In addition, we provide an experimental test by unveiling a distal-to-proximal gradient of Hoxd13 in the vertebrate developing limb bud driven by cell lineage transport.
W. Driever and C. Nusslein-Volhard, Cell 54 , 95-104 (1988).
L. Wolpert, J. Theor. Biol. 25 , 1-47 (1969).
J.B. Gurdon, and P.Y. Bourillot, Nature 413 , 797-803 (2001).
A.D. Lander, Q. Nie, and F.Y. Wan, Dev. Cell 2 , 785-796 (2002).
K. Kruse, P. Pantazis, T. Bollenbach, F. Jülicher and M. Gonzalez-Gaitan, Development 131 , 4843-56 (2004).
A.J. Zhu, and M.P. Scott, Genes Dev. 18 , 2985-2997 (2004).
S. Pfeiffer, C. Alexandre, M. Calleja and J.P. Vincent, Curr. Biol. 10 , 321-324 (2000).
S.J. Gaunt, D. Drage and A. Cockley, Mech. Dev. 120 , 573-86 (2003).
J. Dubrulle, and O. Pourquié, Nature 427 , 419-422 (2004).
I. Golding, J. Paulsson, S.M. Zawilski and E.C. Cox, Cell 123 , 1025-1036 (2005).
Force dependent fragility in RNA hairpins
1 Departament de Fisica Fonamental, Universitat de Barcelona, Spain.
2 Merck & Co. Inc., Automated Biotechnology Dpt., North Wales PA 19454, USA
We apply Kramers theory for the dissociation of multiple bonds under mechanical force to interpret experimental results for the unfolding/refolding force distributions of RNA hairpins pulled at different loading rates using laser tweezers. We identify two different kinetic regimes depending on the range of forces explored during the unfolding and refolding process. The present approach extends the range of validity of the two-states approximation by providing a theoretical framework to reconstruct free-energy profiles and identify force-dependent kinetic barriers along reaction coordinates in single molecule pulling experiments. The method should be applicable to RNA hairpins with multiple kinetic barriers.
Intrinsic fluctuations, robustness and tunability in signaling cycles
1 Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA
2 Harvard-MIT Division of Health Science and Technology & Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Covalent modification cycles (e.g. phosphorylation) underlie most cellular signaling. Low molecular copy number, arising from compartmental segregation and slow diffusion between compartments, potentially renders these cycles vulnerable to intrinsic chemical fluctuations. How can a cell operate reliably in the presence of this inherent stochasticity? How do changes in extrinsic parameters lead to variability of response? Can cells exploit these parameters to tune cycles to different ranges of stimuli?
We study the dynamics of an isolated phosphorylation cycle. Our model shows that the cycle transmits information reliably if it is tuned to an optimal parameter range, in spite of intrinsic fluctuations and even for small input signal amplitudes. At the same time, the cycle is sensitive to changes in the concentration and activity of kinases and phosphatases. This sensitivity can lead to significant cell-to-cell response variability
Our results show that signaling cycles possess a surprising combination of robustness and tunability. This combination makes them ubiquitous in eukaryotic signaling, optimizing signaling in the presence of fluctuations using their inherent flexibility. On the other hand, cycles tuned to suppress intrinsic fluctuations can be fragile to changes in the number and activity of kinases and phosphatases. Such trade-offs in robustness to fluctuations can influence the evolution of signaling cascades, making them the weakest links in cellular circuits.
The precision of genetic oscillators and clocks
Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
Genetic oscillations are known to play a major role in driving different cellular processes, for example in circadian clocks, during the cell cycle, and for patterning the vertebrate body axis. Due to the stochastic nature of gene expression, the period of these oscillations is subject to fluctuations. These fluctuations limit the precision of the oscillator, which can be quantified by the quality factor. We study the precision of genetic oscillators in a simple but general stochastic feedback system. We show that a high quality factor is possible even when the number of molecules is low and amplitude fluctuations are large. We relate our results to circadian clocks in bacteria, where high quality oscillations have been observed in single cell experiments, with a precision of more than 100 days.
Frequency encoding, futile cycling and identification of feedbacks in Ca2+/IP3 signalling
1 Humboldt University at Berlin, Germany
2 University of Medicine and Dentistry of New Jersey, USA
Hormones that act through the calcium-releasing messenger, inositol 1,4,5-trisphosphate (IP(3)), cause intracellular calcium oscillations, which have been ascribed to calcium feedbacks on the IP(3) receptor. Recent studies have shown that IP(3) levels oscillate together with the cytoplasmic calcium concentration. To investigate the functional significance of this phenomenon, we have developed mathematical models of the interaction of both second messengers. The models account for both positive and negative feedbacks of calcium on IP(3) metabolism, mediated by calcium activation of phospholipase C and IP(3) 3-kinase, respectively. The coupled IP(3) and calcium oscillations have a greatly expanded frequency range compared to calcium fluctuations obtained with clamped IP(3). Therefore the feedbacks can be physiologically important in supporting the efficient frequency encoding of hormone concentration observed in many cell types. This action of the feedbacks depends on the turn-over rate of IP(3). To shape the oscillations, positive feedback requires fast IP(3) turnover, whereas negative feedback requires slow IP(3) turnover. The ectopic expression of an IP(3) binding protein has been used to decrease the rate of IP(3) turnover experimentally, resulting in a dose-dependent slowing and eventual quenching of the Ca(2+) oscillations. These results are consistent with a model based on positive feedback of Ca(2+) on IP(3) production.
Robust stem cell fate control in Drosophila
Spanish National Cancer Centre, Madrid, Spain
Stem cells are the essential precursors of all cell types in our bodies. A particular tissue location, known as niche, provides the necessary factors for their maintenance, i.e., stem cell self-renewal [1]. How does a stem cell decide to abandon the niche and initiate the program of differentiation? Here, I will present the system–level features of a control module regulating this decision [2]. First, I will introduce the molecular players and interactions associated to stem cells of the Drosophila ovary niche. Then, I will discuss how these molecular constituents establish a robust two-layer control module regulating stem cell differentiation. At the intracellular level, the combination of various positive and negative feedback loops enhances the robustness of stem cell fate commitment even under the inherently noisy cellular milieu. In addition, at the intercellular level, this control structure induces the phenomenon of cell competition among stem cells. Cell competition emerges when cells with unequal levels of protein synthesis meet leading to the death of the “losers” cells. In the niche, competition leads instead to differentiation acting effectively as second extra layer of control. A better knowledge of the mechanisms regulating stem cell self–renewal and differentiation opens unforeseen avenues for understanding the factors contributing to tissue homeostasis and for the development of new biomedical strategies.
H. Lin The stem–cell niche theory: lessons from flies, Nature Review Genetics 3 , 931 (2002).
B. Diaz, I. Fernandez-Ruiz, J.F. Poyatos, E. Moreno, “Stem cell fate control through programmed cell competition”, submitted (2005).
A multiscale modelling framework for multi-cellular systems in Synthetic Biology
Department of Bioengineering, Imperial College London, London, United Kingdom
Synthetic Biology provides an exciting biological paradigm to explore stochastic behaviors in cellular systems. As synthetic systems are moving to more complex and interactive designs, we aim at providing a modelling framework in order to describe a petri dish-like experiment from gene expression to cell population behavior. This framework explores emergent properties of genetic networks throughout the biological levels.
Current synthetic biology systems, implemented in E.Coli, may be described at 3 different levels: the synthetic genetic network, the host and the population of hosts. In our framework, we use Stochastic Petri Nets (SPN) to model the genetic network dynamic [1]. Petri Nets allow a natural graphical representation for design and visualization purposes. They are also an optimized structure to handle efficient hybrid stochastic simulations.
We also use a cell-agent to encapsulate the SPN, as E.Coli does for synthetic plasmids. The agent enables phenotypic behaviours such as division, apoptosis, continuous migration, secretion and uptake. These behaviours can be triggered by the simulated SPN or modeled as independent stochastic events. Finally, a full agent-based system is formed by all the individual agents immersed into a diffusive environment in order to model quorum sensing like mechanism.
This multiscale approach provides a simulation platform with a high degree of compound traceability. We have also implemented specific filters in order to export simulated data into experimental data format such as the Flow Cytometry Standard (FCS3.0) and Open Microscopy Environment (OME). By doing so, we wish to ease the validation process when it comes to compare results with real experiments. It is also intended to define a common interface with experimentalists in order to guide new experimental or simulation settings. We illustrate the use of our framework on the synthetic system defined by You et al [2].
Peter J. E. Goss and Jean Peccoud, “Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets'', Biochemistry 95 , 6750 (1998).
L. You, R. S. Cox III, R. Weiss, F. H. Arnold, “Programmed population control by cell-cell communication and regulated killing”, Nature 428 , 868 (2004).
Stochastic dynamics of macromolecular-assembly networks
Integrative Biological Modeling Laboratory, Computational Biology Program, Memorial Sloan-Kettering Cancer Center, 1275 York Ave., Box #460, New York, USA
The formation and regulation of macromolecular assemblies provides the backbone of most cellular processes, including gene regulation and signal transduction. The inherent complexity of assembling macromolecular structures makes current computational methods strongly limited for understanding how the physical interactions between cellular components give rise to systemic properties of cells. Here we present a stochastic approach to study the dynamics of networks formed by macromolecular complexes in terms of the molecular interactions of their components [1]. Exploiting key thermodynamic concepts, this approach makes it possible to both estimate reaction rates and incorporate the resulting assembly dynamics into the stochastic kinetics of cellular networks. As prototype systems, we consider the lac operon and phage-λ induction switches, which rely on the formation of DNA loops by proteins [2] and on the integration of these protein-DNA complexes into intracellular networks. This cross-scale approach offers an effective starting point to move forward from network diagrams, such as those of protein-protein and DNA-protein interaction networks, to the actual dynamics of cellular processes. Special attention will be paid to discuss the role of DNA looping as a cellular built-in mechanism to control both the intrinsic fluctuations of transcription and cell-to-cell variability [1,3].
L. Saiz and J. M. G. Vilar, ``Stochastic dynamics of macromolecular-assembly networks'', Molecular Systems Biology, doi:10.1038/msb4100061 (2006).
L. Saiz and J. M. G. Vilar, ``DNA looping: the consequences and its control'', Current Opinion in Structural Biology 16(3), in press (2006).
J. M. G. Vilar and L. Saiz, ``DNA looping in gene regulation: From the assembly of macromolecular complexes to the control of transcriptional noise'', Current Opinion in Genetics and Development 15, 136 (2005).
Min protein oscillations and segregation during cell division
Department of Mathematics, Imperial College London, London, United Kingdom
The Min system in Escherichia coli directs division to the centre of the cell through pole-to-pole oscillations of the MinCDE proteins. MinC and MinD first accumlate in one half of the cell preventing formation of the septal ring in this region. MinE forms a ring near midcell which moves towards the pole, displacing MinC and MinD. MinC and MinD then gather at the opposite pole and the process repeats. Although the reaction steps involved in the oscillation cycle have been studied, the behaviour of the Min system during cell division itself has not been investigated in detail. In this talk I will present a one dimensional stochastic model of the Min system which incorporates membrane polymerisation of MinD into linear chains. This model reproduces much of the observed phenomenology, including pole-to-pole oscillations of the Min proteins. We then apply this model to investigate the Min system during cell division. Oscillations continue initially unaffected by the closing septum, before cutting off rapidly. The fractions of Min proteins in the daughter cells vary widely, from 50%-50% up to 85%-15% of the total from the parent cell, suggesting that there may be another mechanism for regulating these levels in vivo.
1. F. Tostevin and M. Howard, Phys. Biol. 2006 3 :1–12.
Dynamics of synthetic genetic networks with repressive cell-to-cell communication
1 Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany
2 University of Potsdam, Germany
3 Dept. de Física i Enginyeria Nuclear, Univ. Politècnica de Catalunya, Terrassa, Spain
4 Department Theoretical Physics, Lebedev Physical Inst., Russia
Natural genetic circuits are inextricably embedded within the complex genetic network that underlies the full cellular machinery. This complicates considerably the study of dynamics of isolated genetic modules, which is currently possible almost exclusively with synthetic genetic circuits. These systems consist of a limited number of genes that are designed to operate relatively isolated from the rest of the cellular machinery. Thus, synthetic genetic networks provide a relatively well controlled test system in which the functions of natural gene networks can be isolated and characterized in detail. We suggest a modification of the repressilator model [1] with quorum sensing [2] for the intercell coupling to realize a repressive cell-to-cell communication instead of the activatory coupling. Numerical simulations in an experimental realistic parameter range have shown a very rich dynamics and revealed e.g. the formation of clusters oscillating with a phase slip to other clusters, the oscillation death, completely synchronized oscillations and suppression of any oscillations. Different dynamical regimes can be expected that are highly relevant for biological systems. For example the oscillating clusters with a phase slip could be a protection mechanism of the cell culture to environmental stress and the formation of clusters of steady state behavior (oscillation death), could be a manifestation of cell diversification and differentiation. Some of these dynamical regimes coexist in parallel at a given parameter set (environmental condition) and hence the initial conditions determine the resulting dynamical behavior in this situation, i.e. depending on the history of the cell culture it shows different behaviors.
M.B. Elowitz and S. Leibler, “A synthetic oscillatory network of transcriptional regulators”, Nature 403 , 335 (2000).
Jordi García-Ojalvo and Michael B. Elowitz and Steven H. Strogatz, “Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing”, Proc. Natl. Acad. Sci. U.S.A. 101 , 10955 (2004).
Interspike interval densities in resonate and fire neurons
Institute for Physics, Humboldt University at Berlin, Germany
The subthreshold dynamics of a neural cell can follow one of the two patterns: the resonant neurons generate intrinsic subthreshold membrane potential oscillations, while in nonresonant neurons these oscillations can not be observed. We investigate how do these subthreshold behaviors affect the suprathreshold response. The dynamics of neurons is given by a resonate and fire model with experimentally obtained parameter values.
We start from an exact expression for the first passage time density in terms of an infinite series of integrals over joint densities of level crossings. For rapidly decaying correlations closed analytic expressions can be obtained based on approximate summation of this series. We use these analytic expressions for a linear oscillator model with threshold and reset and determine the multipeaked interspike interval density. We show that a change in model parameters induces qualitative changes in the interspike interval densities.
T. Verechtchaguina, I.M. Sokolov, and L. Schimansky-Geier, Europhys. Lett., submitted (2006).
Stochastic and spatial modelling of biochemical networks
Centre for Mathematics and Computer Science, Universiteit van Amsterdam, The Netherlands
An important topic in biochemical modelling are models capable of capturing spatio-temporal phenomena as well as stochastic effects due to molecular fluctuations. An important aim in the these models is to include spatial cellular structures. This requires additional levels of details to be included in the model. We focus on stochastic and spatial phenomena affecting membrane-cytosol reactions, which in some cases give rise to gradients, or induce waves in the cytosol.
We have developed a particle-based solver for the Reaction-Diffusion Master Equation, which consist of a discretisation of space and separates the reaction and diffusion operators. At the core of the method lie Gillespie's method for reaction and a multiparticle diffusion method. In the multi-particle method, the size of the lattice constrains both the accuracy and the efficiency of the simulation. Spatial localised reactions, such as membrane-bound reactions, introduce new constraints on the lattice size and on how such reactions are treated. Additionally, different parts of the pathway can operate in different levels of detail. This means, that while in some cases a macroscopic simulation might suffice, in other cases a microscopic simulation will be required. In our model we have applied a multiple grid approach to solve these issues.
We will discuss elementary properties of Reaction-Diffusion Master Equation methods and show an application of our method in a biological case study and discuss requirements for simulations of realistic biochemical processes with spatial geometries.
Noise-induced frequency locking in a simple genetic oscillator
1 Nonlinear Dynamics and Chaos Group, Universidad Rey Juan Carlos, Madrid, Spain
2 Department of Electrical Eng. and Electronics, Osaka Sangyo University, Osaka, Japan
3 ERATO Aihara Complexity Modelling Project, Institute of Industrial Science, University of Tokyo, Tokyo, Japan
Cell cycle plays a fundamental role in all living beings. The constant oscillation of the cell division is the way by which the cell reproduces itself and contributes to the development of the organism. This oscillatory process can have a significant influence on other biological functions. We propose to focus on the influence of the cell cycle on a simple genetic oscillator, which is composed of a simple feedback loop with delay. This system shows robust oscillations for a large range of time delays. Without direct coupling of the genetic network, we show that the cell cycle is able to entrain the genetic oscillator very quickly. Furthermore, when the noise in the system is increased the frequency locking is improved. As a matter of fact, the range of parameters for which the frequency locking arises is larger.
1 University of Potsdam, Germany
2 Dept. de Física i Enginyeria Nuclear, Univ. Politècnica de Catalunya,Terrassa, Spain
Due to the complexity of underlying processes the study of gene expression dynamics it is easier to study it with synthetic genetic networks, which include limited numbers of interacting genes. One of experimentally designed synthetic genetic oscillators is a relaxator which is constructed by combining two gene networks, one of which is a toggle switch, in which two genes mutually repress each other, The second network is based on the dynamics of an autoinducer which drives the toggle switch and provides an intercell communication via quorum sensing. The dynamics of this systems is very rich and demonstrate different dynamical regimes from synchronization to oscillation death and becomes even more complicated if the gene expression is stochastic. We simulate numerically a system of coupled noisy genetic relaxators and investigate how dynamical regimes can be controlled by noise, quorum-sensing, population size and cell growth. We show that noise can switch between different dynamics in this system. We have also shown that in a system of coupled noisy oscillators the dynamical regimes can be controlled via cell growth. Noise plays a crucial role in this effect via system size effect mechanism. In this way one can program the desired dynamical regime up from a certain population size. This can contribute to understanding of cell differentiation mechanisms. Further on, we show that noise or stochasticity despite its random input can reduce dynamics and induce oscillation death via formation of one cluster.