Modelling protein localisation and positional information in subcellular systems

Cells and their component structures are highly organised. The correct function of many biological systems relies upon not only temporal control of protein levels but also spatial control of protein localisation within cells. Mathematical modelling allows us to quantitatively test potential mechanisms for protein localisation and spatial organisation. Here we present models of three examples of spatial organisation within individual cells. In the bacterium E. coli, the site of cell division is partly determined by the Min proteins. The Min proteins oscillate between the cell poles and suppress formation of the division ring here, thereby restricting division to midcell. We present a stochastic model of the Min protein dynamics, and use this model to investigate partitioning of the Min proteins between the daughter cells during cell division. The Min proteins determine the correct position for cell division by forming a timeaveraged concentration gradient which is minimal at midcell. Concentration gradients are involved in a range of subcellular processes, and are particularly important for obtaining positional information. By analysing the low copy number spatiotemporal uctuations in protein concentrations for a single polar gradient and two oppositelydirected gradients, we estimate the positional precision that can be achieved in vivo. We nd that time-averaging is vital for high precision. The embryo of the nematode C. elegans has become a model system for the study of cell polarity. At the one-cell stage, the PAR proteins form anterior and posterior domains in a dynamic process driven by contraction of cortical actomyosin. We present a continuum model for this system, including a highly simpli ed model of the actomyosin dynamics. Our model suggests that the known PAR protein interactions 5 are insu cient to explain the experimentally observed cytoplasmic polarity. We discuss a number of modi cations to the model which reproduce the correct cytoplasmic distributions

Fundamental Limits to Position Determination by Concentration Gradients

Position determination in biological systems is often achieved through protein concentration gradients. Measuring the local concentration of such a protein with a spatially-varying distribution allows the measurement of position within the system. In order for these systems to work effectively, position determination must be robust to noise. Here, we calculate fundamental limits to the precision of position determination by concentration gradients due to unavoidable biochemical noise perturbing the gradients. We focus on gradient proteins with first order reaction kinetics. Systems of this type have been experimentally characterised in both developmental and cell biology settings. For a single gradient we show that, through time-averaging, great precision can potentially be achieved even with very low protein copy numbers. As a second example, we investigate the ability of a system with oppositely directed gradients to find its centre. With this mechanism, positional precision close to the centre improves more slowly with increasing averaging time, and so longer averaging times or higher copy numbers are required for high precision. For both single and double gradients, we demonstrate the existence of optimal length scales for the gradients, where precision is maximized, as well as analyzing how precision depends on the size of the concentration measuring apparatus. Our results provide fundamental constraints on the positional precision supplied by concentration gradients in various contexts, including both in developmental biology and also within a single cell.Comment: 24 pages, 2 figure

Multiplexing Biochemical Signals

In this paper we show that living cells can multiplex biochemical signals, i.e. transmit multiple signals through the same signaling pathway simultaneously, and yet respond to them very specifically. We demonstrate how two binary input signals can be encoded in the concentration of a common signaling protein, which is then decoded such that each of the two output signals provides reliable information about one corresponding input. Under biologically relevant conditions the network can reach the maximum amount of information that can be transmitted, which is 2 bits.Comment: 4 pages, 4 figure

A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division

The Min system in Escherichia coli directs division to the centre of the cell through pole-to-pole oscillations of the MinCDE proteins. We present a one dimensional stochastic model of these oscillations which incorporates membrane polymerisation of MinD into linear chains. This model reproduces much of the observed phenomenology of the Min system, including pole-to-pole oscillations of the Min proteins. We then apply this model to investigate the Min system during cell division. Oscillations continue initially unaffected by the closing septum, before cutting off rapidly. The fractions of Min proteins in the daughter cells vary widely, from 50%-50% up to 85%-15% of the total from the parent cell, suggesting that there may be another mechanism for regulating these levels in vivo.Comment: 19 pages, 12 figures (25 figure files); published at http://www.iop.org/EJ/journal/physbi

Modelling the Establishment of PAR Protein Polarity in the One-Cell C. elegans Embryo

At the one-cell stage, the C. elegans embryo becomes polarized along the anterior-posterior axis. The PAR proteins form complementary anterior and posterior domains in a dynamic process driven by cytoskeletal rearrangement. Initially, the PAR proteins are uniformly distributed throughout the embryo. Following a cue from fertilization, cortical actomyosin contracts towards the anterior pole. PAR-3/PAR-6/PKC-3 (the anterior PAR proteins) become restricted to the anterior cortex. PAR-1 and PAR-2 (the posterior PAR proteins) become enriched in the posterior cortical region. We present a mathematical model of this polarity establishment process, in which we take a novel approach to combine reaction-diffusion dynamics of the PAR proteins coupled to a simple model of actomyosin contraction. We show that known interactions between the PAR proteins are sufficient to explain many aspects of the observed cortical PAR dynamics in both wild-type and mutant embryos. However, cytoplasmic PAR protein polarity, which is vital for generating daughter cells with distinct molecular components, cannot be properly explained within such a framework. We therefore consider additional mechanisms that can reproduce the proper cytoplasmic polarity. In particular we predict that cytoskeletal asymmetry in the cytoplasm, in addition to the cortical actomyosin asymmetry, is a critical determinant of PAR protein localization.Comment: 28 pages, 8 figure