Quantitative Biology

p27 Kip1 (p27) is an intrinsically disordered protein (IDP) that folds upon binding to cyclin-dependent kinase (Cdk) / cyclin complexes (e.g., Cdk2 / cyclin A), inhibiting their catalytic activity and causing cell cycle arrest. However, cell division progresses when stably Cdk2 / cyclin A-bound p27 is phosphorylated on one or two structurally occluded tyrosine residues [ tyrosines 88 (Y88) and 74 (Y74) ] and a distal threonine residue [ threonine 187 (T187) ] . These events trigger ubiquitination and degradation of p27, fully activating Cdk2 / cyclin A to drive cell division. Using an integrated approach comprising structural, biochemical, biophysical and single-molecule fluorescence methods, we show that Cdk2 / cyclin A-bound p27 samples lowly-populated conformations that dynamically anticipate the sequential steps of this signaling cascade. "Dynamic anticipation" provides access to the non-receptor tyrosine kinases, BCR-ABL and Src, which sequentially phosphorylate Y88 and Y74 and promote intra-assembly phosphorylation (of p27) on distal T187. Tyrosine phosphorylation also allosterically relieves p27-dependent inhibition of substrate binding to Cdk2 / cyclin A, a phenomenon we term "cross-complex allostery". Even when tightly bound to Cdk2 / cyclin A, intrinsic flexibility enables p27 to integrate and process signaling inputs, and generate outputs including altered Cdk2 activity, p27 stability, and, ultimately, cell cycle progression. Intrinsic dynamics within multi-component assemblies may be a general mechanism of signaling by regulatory IDPs, which can be subverted in human disease, as exemplified by hyper-active BCR-ABL and Src in certain cancers.

Dynamic instability of microtubules is considered using frameworks of non-linear thermodynamics and non-equilibrium reaction-diffusion systems. Stochastic assembly/disassembly phases in the polymerization dynamics of microtubules are treated as a result of collective clusterization of microdefects (holes in structure). The model explains experimentally observed power law dependence of catastrophe frequency from the microtubule growth rate. Additional reaction-diffusion-precipitation model is developed to account for kinetic limitations in microtubule dynamics. It is shown that large scale periodic microtubules length fluctuations are accompanied by concentration autowaves. We built corresponding parametric diagram mapping areas of stationary, non-stationary and metastable solutions. The loss of stability for the stationary solutions happens through bifurcation of Andronov-Hopf. Using parametric diagram we classify cytostatic effect of microtubule stabilizing drugs in four major classes and analyze their compatibility and possible synergistic effect for cancer treatment therapy.

The cell nucleus houses the chromosomes, which are linked to a soft shell of lamin filaments. Experiments indicate that correlated chromosome dynamics and nuclear shape fluctuations arise from motor activity. To identify the physical mechanisms, we develop a model of an active, crosslinked Rouse chain bound to a polymeric shell. System-sized correlated motions occur but require both motor activity {\it and} crosslinks. Contractile motors, in particular, enhance chromosome dynamics by driving anomalous density fluctuations. Nuclear shape fluctuations depend on motor strength, crosslinking, and chromosome-lamina binding. Therefore, complex chromatin dynamics and nuclear shape emerge from a minimal, active chromosome-lamina system.

We investigate the microtubule polymerization dynamics with catastrophe and rescue events for three different confinement scenarios, which mimic typical cellular environments: (i) The microtubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it grows against an elastic obstacle with a linearly increasing force. We use realistic catastrophe models and analyze the microtubule dynamics, the resulting microtubule length distributions, and force generation by stochastic and mean field calculations; in addition, we perform stochastic simulations. We also investigate the force dynamics if growth parameters are perturbed in dilution experiments. Finally, we show the robustness of our results against changes of catastrophe models and load distribution factors.

We study the dynamics of intracellular calcium oscillations in the presence of proteins that bind calcium on multiple sites and that are generally believed to act as passive calcium buffers in cells. We find that multisite calcium-binding proteins set a sharp threshold for calcium oscillations. Even with high concentrations of calcium-binding proteins, internal noise, which shows up spontaneously in cells in the process of calcium wave formation, can lead to self-oscillations. This produces oscillatory behaviors strikingly similar to those observed in real cells. In addition, for given intracellular concentrations of both calcium and calcium-binding proteins the regularity of these oscillations changes and reaches a maximum as a function noise variance, and the overall system dynamics displays stochastic coherence. We conclude that calcium-binding proteins may have an important and active role in cellular communication.

We present the mechanism of interaction of Wnt network module, which is responsible for periodic sometogenesis, with p53 regulatory network, which is one of the main regulators of various cellular functions, and switching of various oscillating states by investigating p53-Wnt model. The variation in Nutlin concentration in p53 regulating network drives the Wnt network module to different states, stabilized, damped and sustain oscillation states, and even to cycle arrest. Similarly, the change in Axin concentration in Wnt could able to modulate the p53 dynamics at these states. We then solve the set of coupled ordinary differential equations of the model using quasi steady state approximation. We, further, demonstrate the change of p53 and Gsk3 interaction rate, due to hypothetical catalytic reaction or external stimuli, can able to regulate the dynamics of the two network modules, and even can control their dynamics to protect the system from cycle arrest (apoptosis).

Receptor-mediated endocytosis requires that the energy of adhesion overcomes the deformation energy of the plasma membrane. The resulting driving force is balanced by dissipative forces, leading to deterministic dynamical equations. While the shape of the free membrane does not play an important role for tensed and loose membranes, in the intermediate regime it leads to an important energy barrier. Here we show that this barrier is similar to but different from an effective line tension and suggest a simple analytical approximation for it. We then explore the rich dynamics of uptake for particles of different shapes and present the corresponding dynamical state diagrams. We also extend our model to include stochastic fluctuations, which facilitate uptake and lead to corresponding changes in the phase diagrams.

A classical example of a mathematical model for oscillations in a biological system is the Selkov oscillator, which is a simple description of glycolysis. It is a system of two ordinary differential equations which, when expressed in dimensionless variables, depends on two parameters. Surprisingly it appears that no complete rigorous analysis of the dynamics of this model has ever been given. In this paper several properties of the dynamics of solutions of the model are established. With a view to studying unbounded solutions a thorough analysis of the Poincaré compactification of the system is given. It is proved that for any values of the parameters there are solutions which tend to infinity at late times and are eventually monotone. It is shown that when the unique steady state is stable any bounded solution converges to the steady state at late times. When the steady state is unstable it is shown that for given values of the parameters either there is a unique periodic solution to which all bounded solutions other than the steady state converge at late times or there is no periodic solution and all solutions other than the steady state are unbounded. In the latter case each unbounded solution which tends to infinity is eventually monotone and each unbounded solution which does not tend to infinity has the property that each variable takes on arbitrarily large and small values at arbitrarily late times.

The bacterial flagellar motor drives the rotation of flagellar filaments and enables many species of bacteria to swim. Torque is generated by interaction of stator units, anchored to the peptidoglycan cell wall, with the rotor. Recent experiments [Yuan, J. & Berg, H. C. (2008) PNAS 105, 1182-1185] show that near zero load the speed of the motor is independent of the number of stators. Here, we introduce a mathematical model of the motor dynamics that explains this behavior based on a general assumption that the stepping rate of a stator depends on the torque exerted by the stator on the rotor. We find that the motor dynamics can be characterized by two time scales: the moving-time interval for the mechanical rotation of the rotor and the waiting-time interval determined by the chemical transitions of the stators. We show that these two time scales depend differently on the load, and that their crossover provides the microscopic explanation for the existence of two regimes in the torque-speed curves observed experimentally. We also analyze the speed fluctuation for a single motor using our model. We show that the motion is smoothed by having more stator units. However, the mechanism for such fluctuation reduction is different depending on the load. We predict that the speed fluctuation is determined by the number of steps per revolution only at low load and is controlled by external noise for high load. Our model can be generalized to study other molecular motor systems with multiple power-generating units.

In this supporting information we briefly describe the torque-speed measurement procedure. We show the hook spring compliance used in the simulations. We analyze the distribution functions of the moving and waiting time intervals. We study the dependence of the torque plateau region on the stator jumping rate and the cutoff angle, and the robustness of the results against different rotor-stator- potential and load-rotor forces.