New Methods for Chaotic Dynamics

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In this case, the model has the following additional equations, where P 1 and P 2 represent the concentrations of the two proteins and C 2,1 the concentration of the complex:. An exploitation of the effects of the entire parameter space will be interesting to pursue in future work, but is beyond the scope of this paper. Obviously if the two subunits are both HAG proteins, the complex has the highest average level in the oscillatory regime, while if it consists of two LAG proteins, the highest average level will be found in the chaotic regime Fig.

Simulating the above equations for a heterogenous complex, we find a significantly higher level of the complex in the chaotic regime, as seen in Fig. Effects of chaos on protein complex formation. Red: Level of HAG protein. Blue: Level of LAG protein multiplied by The y -axis show the concentration in chaos divided by the concentration in the oscillatory regime, and the black line show where these are equal. Same axis as in g. We then test larger complexes. As we see in Fig. Unexpectedly, we find that all heterogeneous complexes exhibit a higher average lvel in the chaotic regime Fig.

This means that only homogenous HAG complexes would be present at a high level in the single-mode oscillatory regime. As seen in Fig.

Chaotic economic dynamics

Another economical argument for the cell is that if only the complexes are of importance, then it is necessary to minimise the number of unused subunits. This ratio too is largest in the chaotic regime for all complexes, except those made only from HAG proteins. Thus, a chaotically varying transcription factor not only up regulates LAGs, but also results in higher and more economical production of protein complexes composed of subunits from different genes.

In all simulations, cells have randomly distributed initial conditions, i. Within each cell, we will track one LAG and one HAG; parameters are chosen so that the two corresponding proteins have the same average protein level.

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In the chaotic regime, the distribution is broad and heterogenous for both proteins Fig. Population heterogeneity emerges from chaos. Bottom: The concentration corresponding to a single-mode oscillation; TNF amplitude: 0.

Middle: The concentration corresponding to mode-hopping; TNF amplitude: 0. Top: The concentration corresponding to chaos; TNF amplitude: 0. TNF amplitudes are identical to those used in a. TNF amplitudes are identical to those used in c.

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The panel below shows a specific trajectory on this pattern. Such heterogeneity in a cell population can provide a selective advantage when the population is exposed to some potentially lethal stresses. Imagine each cell in the population is exposed to two toxic drugs at concentrations D 1 and D 2.

We assume that at each time step each cell is killed with probability. This describes a situation where the two proteins P 1 and P 2 are stress-responders that can help the cell survive stressed conditions. First we consider the situation where only one of these drugs is present, shown in Fig. This is what one would expect from Fig. When only Drug 2 is added in a high amount, cells in chaotic states will have a slightly higher survival rate, but due to large fluctuations, these cells will also eventually die due to temporary low levels of Protein 2.

Now we consider what will happen to the system if both drugs are added in a comparable amounts. We test four different patterns of adding the drugs Fig. From this we conclude that in the presence of multiple toxic drugs, a population of cells is better off having a large heterogeneity in gene expression and up regulating the LAGs and thus up regulating the product of genes. Transcription factors are known to have different dynamics, depending on external conditions, but how this may be exploited to differentially control downstream genes is not well understood.

We have shown how dynamically varying transcription factors can differentially regulate genes based on an effective affinity that characterises the interaction between the gene and the transcription factor.

In particular, we suggest that chaotic dynamics can produce differential control of high vs. LAGs, down regulating the former while simultaneously up regulating the latter.


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We show that this can be used not only to control single non-interacting genes, but also for upregulating specific complexes of proteins and generating useful heterogeneities in cell populations. Our model has already successfully predicted the existence of mode-hopping for a range of TNF amplitudes However, an experimental realisation of our model 37 , 38 would necessarily be subject to various sources of noise and stochasticity, and it is not obvious that deterministically chaotic behaviour can be practically discerned in the presence of such fluctuations.

Fortunately, many sophisticated methods exist that allow chaos to be distinguished from noise without requiring unreasonably long time series; see for example refs. Chaotic dynamics has thus far been underestimated as a means for controlling genes, perhaps because of its unpredictability. Our work shows that deterministic chaos potentially expands the toolbox available for single cells to control gene expression dynamically and specifically.

We hope this will inspire theoretical and experimental exploration of the presence and utility of chaos in living cells. All deterministic simulations were performed by numerically integrating the dynamical equations using the Runge—Kutta fourth-order method, and for optimisation reasons, some of the equations were simulated using Euler integration. Whenever Euler integration was used it was tested that it generated similar results as the Runge—Kutta fourth-order method. For all stochastic simulations of NF-kB dynamics, we used the Gillespie algorithm For noise in the external TNF oscillations we used Langevin simulations of the different oscillations.

To find the regions of parameter space that exhibit chaotic dynamics, we first computed the standard deviation in the NF-kB amplitudes from each time series, and found the parameter points at which this grew discontinuously, as we increased the TNF amplitude. Within these regions, we further tested for chaos by calculating the divergence of trajectories that started at almost identical initial points, using deterministic simulations.

Parameter regions where such trajectories diverged exponentially were labelled as regions exhibiting chaos. All computer code is available upon request at heltberg nbi. All the data in this paper, was generated using deterministic and stochastic simulations. All scripts to generate the data are available upon request at heltberg nbi. Hoffmann, A. Science , — Nelson, D. Krishna, S. Natl Acad. USA , — Mengel, B.

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Research on the Phase Space Reconstruction Method of Chaotic Time Series

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Chaotic Dynamics

USA 99 , — Lahav, G. Dynamics of the pMdm2 feedback loop in individual cells. Tay, S. Noise facilitates transcriptional control under dynamic inputs. Cell , — Jensen, M. Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli. FEBS Lett. Arnold, V. Transition to chaos by interaction of resonances in dissipative systems. Circle maps.

A 30 , — Pikovsky, A. Cambridge University Press, Cambridge, Stavans, J. Fixed winding number and the quasiperiodic route to chaos in a convective fluid.

Top Authors

Brown, S. Gwinn, E. Frequency locking, quasiperiodicity, and chaos in extrinsic Ge. Tsai, T. Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Goldbeter, A. Computational approaches to cellular rhythms. Nature , — Woller, A. A mathematical model of the liver circadian clock linking feeding and fasting cycles to clock function. Cell Rep. Mondragon-Palomino, O. Entrainment of a population of synthetic genetic oscillators. Heltberg, M. Cell Syst.

Nonlinear Dynamics: Attractors, Strange and Otherwise

Ashall, L. Pulsatile stimulation determines timing and specificity of NF-B-dependent transcription. Science , Gillespie, D. Exact stochastic simulation of coupled chemical reactions. Maienschein-Cline, M.


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