Non-linear Min protein interactions generate harmonics that signal mid-cell division in Escherichia coli

© 2017 Walsh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The Min protein system creates a dynamic spatial pattern in Escherichia coli cells where the proteins MinD and MinE oscillate from pole to pole. MinD positions MinC, an inhibitor of FtsZ ring formation, contributing to the mid-cell localization of cell division. In this paper, Fourier analysis is used to decompose experimental and model MinD spatial distributions into time-dependent harmonic components. In both experiment and model, the second harmonic component is responsible for producing a mid-cell minimum in MinD concentration. The features of this harmonic are robust in both experiment and model. Fourier analysis reveals a close correspondence between the time-dependent behaviour of the harmonic components in the experimental data and model. Given this, each molecular species in the model was analysed individually. This analysis revealed that membrane-bound MinD dimer shows the mid-cell minimum with the highest contrast when averaged over time, carrying the strongest signal for positioning the cell division ring. This concurs with previous data showing that the MinD dimer binds to MinC inhibiting FtsZ ring formation. These results show that non-linear interactions of Min proteins are essential for producing the mid-cell positioning signal via the generation of second-order harmonic components in the time-dependent spatial protein distribution

Molecular Interactions of the Min Protein System Reproduce Spatiotemporal Patterning in Growing and Dividing Escherichia coli Cells

© 2015 Walsh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Oscillations of the Min protein system are involved in the correct midcell placement of the divisome during Escherichia coli cell division. Based on molecular interactions of the Min system, we formulated a mathematical model that reproduces Min patterning during cell growth and division. Specifically, the increase in the residence time of MinD attached to the membrane as its own concentration increases, is accounted for by dimerisation of membrane- bound MinD and its interaction with MinE. Simulation of this system generates unparalleled correlation between the waveshape of experimental and theoretical MinD distributions, suggesting that the dominant interactions of the physical system have been successfully incorporated into the model. For cells where MinD is fully-labelled with GFP, the model reproduces the stationary localization of MinD-GFP for short cells, followed by oscillations from pole to pole in larger cells, and the transition to the symmetric distribution during cell filamentation. Cells containing a secondary, GFP-labelled MinD display a contrasting pattern. The model is able to account for these differences, including temporary midcell localization just prior to division, by increasing the rate constant controlling MinD ATPase and heterotetramer dissociation. For both experimental conditions, the model can explain how cell division results in an equal distribution of MinD and MinE in the two daughter cells, and accounts for the temperature dependence of the period of Min oscillations. Thus, we show that while other interactions may be present, they are not needed to reproduce the main characteristics of the Min system in vivo

Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications

A Systematic Approach to Delay Functions

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions

Patterning of the MinD cell division protein in cells of arbitrary shape can be predicted using a heuristic dispersion relation

© 2016, Paul M. G. Curmi, et al. Many important cellular processes require the accurate positioning of subcellular structures. Underpinning many of these are protein systems that spontaneously generate spatiotemporal patterns. In some cases, these systems can be described by non-linear reaction-diffusion equations, however, a full description of such equations is rarely available. A well-studied patterning system is the Min protein system that underpins the positioning of the FtsZ contractile ring during cell division in Escherichia coli. Using a coordinate-free linear stability analysis, the reaction terms can be separated from the geometry of a cell. The reaction terms produce a dispersion relation that can be used to predict patterning on any cell shape and size. Applying linear stability analysis to an accurate mathematical model of the Min system shows that while it correctly predicts the onset of patterning, the dispersion relation fails to predict oscillations and quantitative mode transitions. However, we show that data from full solutions of the Min model can be used to generate a heuristic dispersion relation. We show that this heuristic dispersion relation can be used to approximate the Min protein patterning obtained by full simulations of the non-linear reaction-diffusion equations. Moreover, it also predicts Min patterning obtained from experiments where the shapes of E. coli cells have been deformed into rectangles or arbitrary shapes. Using this procedure, it should be possible to generate heuristic dispersion relations from protein patterning data or simulations for any patterning process and subsequently use these to predict patterning for arbitrary cell shapes

Motility of an autonomous protein-based artificial motor that operates via a burnt-bridge principle

Inspired by biology, great progress has been made in creating artificial molecular motors. However, the dream of harnessing proteins – the building blocks selected by nature – to design autonomous motors has so far remained elusive. Here we report the synthesis and characterization of the Lawnmower, an autonomous, protein-based artificial molecular motor comprised of a spherical hub decorated with proteases. Its “burnt-bridge” motion is directed by cleavage of a peptide lawn, promoting motion towards unvisited substrate. We find that Lawnmowers exhibit directional motion with average speeds of up to 80 nm/s, comparable to biological motors. By selectively patterning the peptide lawn on microfabricated tracks, we furthermore show that the Lawnmower is capable of track-guided motion. Our work opens an avenue towards nanotechnology applications of artificial protein motors

Time fractional Fisher-KPP and Fitzhugh-Nagumo equations

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times

Generalized fractional power series solutions for fractional differential equations

The extension of fractional power series solutions for linear fractional differential equations with variable coefficients is considered. Generalized series expansions involving integer powers and fractional powers in the independent variable have recently been shown to provide solutions to certain linear fractional order differential equations. In these studies the generalized series could be formulated as the Cauchy product of a standard power series with integer powers and a fractional power series with powers as multiples of a fractional exponent. Here we show that generalized fractional powers series can provide solutions to a wider class of problems if there is no requirement that the series can be formulated as a Cauchy product

Time-fractional geometric Brownian motion from continuous time random walks

We construct a time-fractional geometric Fokker–Planck equation from the diffusion limit of a continuous time random walk with a power law waiting time density, and a biased multiplicative jump length density dependent on the particles’ current position. The bias is related to a force, and an associated potential, through a steady state Boltzmann distribution. The limit of the random walk, with a force derived from a logarithmic potential, defines a stochastic process that is a fractional generalization of geometric Brownian motion. We have investigated the moments for this process. In geometric Brownian motion the expectation of the logarithm of the position of the particle scales linearly with time. In the fractional generalization, this scales as a sub-linear power law in time, similar to anomalous scaling of the mean square displacement in subdiffusion. In financial applications this could be observed as a sub-linear scaling in the logarithmic return

Subdiffusive discrete time random walks via Monte Carlo and subordination

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation. The computation time scales as a power law in the number of time steps with a fractional exponent simply related to the order of the fractional derivative. We also provide an explicit form of a subordinator for discrete time random walks with Sibuya power law waiting times. This subordinator transforms from an operational time, in the expected number of random walk steps, to the physical time, in the number of time steps