Mechanisms for directed transport and organization at subcellular scales

The timely and faithful segregation of genetic material is an essential cellular function that relies on the transport and stable positioning of subcellular components despite the disruptive influence of thermal fluctuations. In prokaryotes, a two-protein system (known as ParABS) has been identified as being responsible for the positioning of low-copy number plasmids and chromosomes prior to cell division. Multiple experimental observations, in vitro reconstitutions and computational modelling efforts support the idea that this system is powered by the ‘burnt-bridge’ Brownian ratchet mechanism. In this thesis we provide computational models that complement these studies to understand how this mechanism generates and sustains directional transport through the transduction of chemical energy into mechanical motion. In particular we study the effects of chemical kinetics, inter-protein interaction strength, system size and availability of proteins that drive this mechanism with an application to the rich protein dynamics observed in vivo. Finally, we simulate a coarse-grained model for a highly polyvalent ‘burnt-bridges’ Brownian ratchet capable of translocating either by rotation or translation and detail the system parameters that govern the transitions between these two distinct modes of motion. The models presented in this thesis provide key insights and make experimentally testable predictions which can be used for the engineering of novel synthetic motor systems

Operational Principles for the Dynamics of the In Vitro ParA-ParB System

In many bacteria the ParA-ParB protein system is responsible for actively segregating DNA during replication. ParB proteins move by interacting with DNA bound ParA-ATP, stimulating their unbinding by catalyzing hydrolysis, that leads to rectified motion due to the creation of a wake of depleted ParA. Recent in vitro experiments have shown that a ParB covered magnetic bead can move with constant speed over a DNA covered substrate that is bound by ParA. It has been suggested that the formation of a gradient in ParA leads to diffusion-ratchet like motion of the ParB bead but how it forms and generates a force is still a matter of exploration. Here we develop a deterministic model for the in vitro ParA-ParB system and show that a ParA gradient can spontaneously form due to any amount of initial spatial noise in bound ParA. The speed of the bead is independent of this noise but depends on the ratio of the range of ParA-ParB force on the bead to that of removal of surface bound ParA by ParB. We find that at a particular ratio the speed attains a maximal value. We also consider ParA rebinding (including cooperativity) and ParA surface diffusion independently as mechanisms for ParA recovery on the surface. Depending on whether the DNA covered surface is undersaturated or saturated with ParA, we find that the bead can accelerate persistently or potentially stall. Our model highlights key requirements of the ParA-ParB driving force that are necessary for directed motion in the in vitro system that may provide insight into the in vivo dynamics of the ParA-ParB system

DNA Segregation Under Par Protein Control

The spatial organization of DNA is mediated by the Par protein system in some bacteria. ParB binds specifically to the parS sequence on DNA and orchestrates its motion by interacting with ParA bound to the nucleoid. In the case of plasmids, a single ParB bound plasmid is observed to execute oscillations between cell poles while multiple plasmids eventually settle at equal distances from each other along the cell’s length. While the potential mechanism underlying the ParA-ParB interaction has been discussed, it remains unclear whether ParB-complex oscillations are stable limit cycles or merely decaying transients to a fixed point. How are dynamics affected by substrate length and the number of complexes? We present a deterministic model for ParA-ParB driven DNA segregation where the transition between stable arrangements and oscillatory behaviour depends only on five parameters: ParB-complex number, substrate length, ParA concentration, ParA hydrolysis rate and the ratio of the lengthscale over which the ParB complex stimulates ParA hydrolysis to the lengthscale over which ParA interacts with the ParB complex. When the system is buffered and the ParA rebinding rate is constant we find that ParB-complex dynamics is independent of substrate length and complex number above a minimum system size. Conversely, when ParA resources are limited, we find that changing substrate length and increasing complex number leads to counteracting mechanisms that can both generate or subdue oscillatory dynamics. We argue that cells may be poised near a critical level of ParA so that they can transition from oscillatory to fixed point dynamics as the cell cycle progresses so that they can both measure their size and faithfully partition their genetic material. Lastly, we show that by modifying the availability of ParA or depletion zone size, we can capture some of the observed differences in ParB-complex positioning between replicating chromosomes in B. subtilis cells and low-copy plasmids in E. coli cells

Dependence of bead speed on <i>ϕ</i> and ParA rebinding kinetics.

<p>(A) Speed of bead versus time for undersaturating (<i>ϕ</i> < 1) and saturating (<i>ϕ</i> > 1) ParA concentrations. When the system is saturated, the bead attains a constant speed. In the undersaturated conditions, the bead shows a period of persistent acceleration. For both cases, non-cooperative rebinding of ParA was used with a rate of <i>k</i>_<i>r</i> = 0.25 and <i>c</i> = 0.5. (B) The dependence of the final speed attained by the bead on the rate of rebinding, <i>k</i>_<i>r</i>. When <i>ϕ</i> ⩾ 1 increasing <i>k</i>_<i>r</i> reduces the final speed of the bead as there is increased ParA recovery in the wake region created by the bead. For <i>ϕ</i> ≲ 1, the final speed of the bead is greater than the case without any rebinding present (<i>k</i>_<i>r</i> = 0) and reduces to a constant value as <i>k</i>_<i>r</i> is increased (inset). (C) Dependence of the bead’s speed on the non-cooperative rebinding rate and <i>ϕ</i>. In the saturating regime (<i>ϕ</i> > 1) the bead can stall if the rebinding rate is sufficiently large (black curve with <i>k</i>_<i>r</i> = 0.25) (D) Inclusion of cooperative rebinding (black curve with <i>k</i>_<i>c</i> = 5.0) can stall the bead at lower values of <i>ϕ</i> compared to non-cooperative rebinding alone (red curve with <i>k</i>_<i>c</i> = 0.0).</p

Rebinding of ParA proteins leads to two dynamical regimes.

<p>(A) Schematic for the model including rebinding. Now the DNA surface is described by a concentration of binding sites for ParA at a given position, <i>d</i>(<i>x</i>). From position to position it varies around an average concentration of sites given by <i>D</i>₀. Free ParA in the buffer, <i>a</i>_<i>b</i> can now bind to locations on the surface that have unoccupied binding sites. (B) Simulated results for the time evolution of the surface bound ParA for <i>ϕ</i> = <i>A</i>_<i>s</i>/<i>D</i>₀ < 1 (<i>ϕ</i> = 0.5) from <i>τ</i> = 60.0 (light blue) to <i>τ</i> = 100.0 (dark blue). The amplitude of the ParA wavefront created by the bead rises with time, causing the bead’s speed to increase. (C) Simulated results of the time evolution of the surface bound ParA for <i>ϕ</i> > 1 (<i>ϕ</i> = 1.1) from <i>τ</i> = 40.0 (light blue) to <i>τ</i> = 70.0 (dark blue). Since the surface is saturated the moving ParA wavefront remains constant in height leading to the bead moving with constant speed. For these simulations only non-cooperative rebinding was considered at a rate <i>k</i>_<i>r</i> = 0.25 and <i>c</i> = 0.5.</p

Surface diffusion of ParA reduces the speed of the bead.

<p>Dependence of the speed of the bead on the ParA protein’s surface diffusion constant, <i>κ</i>. As diffusivity increased the wake created by the bed filled in faster, reducing the total forward force. No ParA rebinding was considered. We used <i>ϕ</i> = 0.8 and <i>c</i> = 0.5.</p

Spatial noise in the initial ParA concentration is sufficient to generate directed motion.

<p>(A) Bead displacement versus time shows an initial lag period where there is no movement. At later times, the bead attains a constant speed as evidenced by the linear increase of displacement with time. The speed is larger for the system with larger <i>A</i>₀. (B) Simulated ParA profiles for a bead starting at <i>x</i>_<i>p</i> = 0 is shown after 450 time steps. The simulation with higher average initial ParA shows greater bead displacement. (C) The speed of the bead is directly proportional to <i>A</i>₀ and does not depend on the magnitude of noise. The error bars for each point are calculated from 100 simulations. The value of <i>c</i> for these simulations is 0.5.</p

Schematic illustration of the ParA-ParB model.

<p>A ParB decorated bead of radius <i>R</i> is attracted by a central force to surface bound ParA at a location, <i>x</i>, with a concentration given by <i>a</i>(<i>x</i>, <i>τ</i>). This force decays with distance from the bead. The position of the bead is given by <i>x</i>_<i>p</i>, and we assume that its speed is proportional to the total force acting on it.</p