31 research outputs found
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
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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