Analysing dynamical behavior of cellular networks via stochastic bifurcations.
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The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation.However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood.Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types.
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Affiliation: Center for Dynamics of Complex Systems, University of Potsdam, Potsdam, Germany. zakharova-as@mail.ru
ABSTRACT
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The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation. However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood. Here, we define a concept of stochastic bifurcations suitable to investigate the dynamical structure of genetic networks, and show that under stochastic influence, the expression of given proteins of interest is defined via the probability distribution of the phase variable, representing one of the genes constituting the system. Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types. |
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Mentions: In the presence of noise, however, a one-to-one correspondence between the deterministic and the stochastic attractors can not be established, since under noise, the lifetime of the attractors is rather short, or they merge. Therefore, it is appropriate to investigate the dynamical changes in the coupled system from the aspect of transformations of the distribution of the phase variable in terms of phenomenological stochastic bifurcations. We demonstrate here two separate cases: i) the system is located left to the tangent bifurcation, , and in the deterministic case only the focus is stable, and ii) , right after the , where the coexistence of five separate attractors is present, two of them are stable (the attractor of the in-phase oscillations and the attractor of the asymmetric oscillations, which is manifested via two separate stable branches - one corresponding to small, and one to large amplitude oscillations). For and a small noise intensity (), the trajectory naturally spends most of the time in the vicinity of the focus and the resulting distribution has one maximum (Fig. 8a). This means that under very small amount of noise, the genetic network produces rather constant protein concentrations, determined by the peak of the probability distribution. An increase of the noise intensity , however, leads to more frequent visits of the trajectory to the region far away from the origin, inducing oscillations in the vicinity of the stable cycles which exist here. Again, a stochastic bifurcation occurs: a transition from a unimodal (for ) to a bimodal distribution () (Fig. 8b). We can state that the increase in influences the dynamical behavior of the genetic network, manifested through changes in the probability for synthesis of a given protein, the in this case. The system has now a complex trajectory, resulting in a possibility that the genetic network expresses different concentration levels, manifested through peaks in the corresponding probability distribution. For however, due to the presence of six separate branches (in the deterministic case), even for small noise intensities, i.e. of the order , a clear multipeak distribution is manifested for the protein concentration of the observed gene (Fig. 8c). The positioning of the peaks in the distribution resembles the stable attractors in the deterministic case: the middle peak of the distribution, e.g., in Fig. 8c corresponds to the positioning of the focus and the stable branch of the small amplitude oscillations. Thus this peak is more pronounced in the corresponding distribution. For increased noise intensity (i.e. ) however, we can not establish any longer direct correspondence to the deterministic attractors. The trajectory which the system performs in the phase plane is again complex, and further leads to the disappearance of the middle peak in the distribution, characteristic for . Thus, a clear, bimodal distribution emerges (Fig. 8d). |
View Article: PubMed Central - PubMed
Affiliation: Center for Dynamics of Complex Systems, University of Potsdam, Potsdam, Germany. zakharova-as@mail.ru