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

Cell-cell communication enhances the capacity of cell ensembles to sense shallow gradients during morphogenesis

Collective cell responses to exogenous cues depend on cell-cell interactions. In principle, these can result in enhanced sensitivity to weak and noisy stimuli. However, this has not yet been shown experimentally, and, little is known about how multicellular signal processing modulates single cell sensitivity to extracellular signaling inputs, including those guiding complex changes in the tissue form and function. Here we explored if cell-cell communication can enhance the ability of cell ensembles to sense and respond to weak gradients of chemotactic cues. Using a combination of experiments with mammary epithelial cells and mathematical modeling, we find that multicellular sensing enables detection of and response to shallow Epidermal Growth Factor (EGF) gradients that are undetectable by single cells. However, the advantage of this type of gradient sensing is limited by the noisiness of the signaling relay, necessary to integrate spatially distributed ligand concentration information. We calculate the fundamental sensory limits imposed by this communication noise and combine them with the experimental data to estimate the effective size of multicellular sensory groups involved in gradient sensing. Functional experiments strongly implicated intercellular communication through gap junctions and calcium release from intracellular stores as mediators of collective gradient sensing. The resulting integrative analysis provides a framework for understanding the advantages and limitations of sensory information processing by relays of chemically coupled cells.Comment: paper + supporting information, total 35 pages, 15 figure

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

General Defocusing Particle Tracking: fundamentals and uncertainty assessment

General Defocusing Particle Tracking (GDPT) is a single-camera, three-dimensional particle tracking method that determines the particle depth positions from the defocusing patterns of the corresponding particle images. GDPT relies on a reference set of experimental particle images which is used to predict the depth position of measured particle images of similar shape. While several implementations of the method are possible, its accuracy is ultimately limited by some intrinsic properties of the acquired data, such as the signal-to-noise ratio, the particle concentration, as well as the characteristics of the defocusing patterns. GDPT has been applied in different fields by different research groups, however, a deeper description and analysis of the method fundamentals has hitherto not been available. In this work, we first identity the fundamental elements that characterize a GDPT measurement. Afterwards, we present a standardized framework based on synthetic images to assess the performance of GDPT implementations in terms of measurement uncertainty and relative number of measured particles. Finally, we provide guidelines to assess the uncertainty of experimental GDPT measurements, where true values are not accessible and additional image aberrations can lead to bias errors. The data were processed using DefocusTracker, an open-source GDPT software. The datasets were created using the synthetic image generator MicroSIG and have been shared in a freely-accessible repository

Indium antimonide crystal growth experiment M562

It was established that ideal diffusion controlled steady state conditions, never accomplished on earth, were achieved during the growth of Te-doped InSb crystals in Skylab. Surface tension effects led to nonwetting conditions under which free surface solidification took place in confined geometry. It was further found that, under forced contact conditions, surface tension effects led to the formation of surface ridges (not previously observed on earth) which isolated the growth system from its container. In addition, it was possible, for the first time, to identify unambiguously: the origin of segregation discontinuities associated with facet growth, the mode of nucleation and propagation of rotational twin boundaries, and the specific effect of mechanical-shock perturbations on segregation. The results obtained prove the advantageous conditions provided by outer space. Thus, fundamental data on solidification thought to be unattainable because of gravity-induced interference on earth are now within reach

Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Exploring the formation of spheroidal galaxies out to z ∼ 1.5 in GOODS

The formation of massive spheroidal galaxies is studied on a visually classified sample extracted from the Advanced Camera for Surveys/Hubble Space Telescope (ACS/HST) images of the Great Observatories Origins Deep Survey north and south fields, covering a total area of 360 arcmin . The sample size (910 galaxies brighter than i = 24) allows us to explore in detail the evolution over a wide range of redshifts (0.4 10 M galaxies by a factor of 2 between z = 1 and 0, in contrast with a factor of ∼50 for lower mass galaxies (10 <M / M <10 ). One-quarter of the whole sample of early types are photometrically classified as blue galaxies. On a volume-limited sample out to z <0.7, the average stellar mass of the blue ellipticals is 5 × 10 M compared to 4 × 10 M for red ellipticals. On a volume-limited subsample out to z = 1.4 probing the brightest galaxies (M <-21), we find the median redshift of blue and red early types: 1.10 and 0.85, respectively. Blue early types only amount to 4 per cent of this sample (compared to 26 per cent in the full sample). The intrinsic colour distribution correlates overall bluer colours with blue cores (positive radial gradients of colour), suggesting an inside-out process of formation. The redshift evolution of the observed colour gradients is incompatible with a significant variation in stellar age within each galaxy. The slope of the Kormendy relation in the subsample of massive galaxies does not change over 0.4 <z <1.4 and is compatible with z = 0 values. The 'zero-point' of the Kormendy relation (i.e. the surface brightness at a fixed half-light radius) is 1 mag fainter (in the B band) for the subsample of low-mass (M <3.5 × 10 M ) early types.Peer reviewe

A determination of the molar gas constant R by acoustic thermometry in helium

We have determined the acoustic and microwave frequencies of a misaligned spherical resonator maintained near the temperature of the triple point of water and filled with helium with carefully characterized molar mass M = (4.002 6032 ± 0.000 0015) g mol-1, with a relative standard uncertainty ur(M) = 0.37×10-6. From these data and traceable thermometry we estimate the speed of sound in our sample of helium at TTPW = 273.16 K and zero pressure to be u0 2 = (945 710.45 ± 0.85) m2 s-2 and correspondingly deduce the value R = (8.314 4743 ± 0.000 0088) J mol-1 K-1 for the molar gas constant. We estimate the value k = R/NA = (1.380 6508 ± 0.000 0015) × 10-23 J K-1 for the Boltzmann constant using the currently accepted value of the Avogadro constant NA. These estimates of R and k, with a relative standard uncertainty of 1.06 × 10-6, are 1.47 parts in 106 above the values recommended by CODATA in 2010

Optical measurement methods in thermogasdynamics

A review is presented of a number of optical methods of flow measurements. Consideration is given to such spectroscopic methods as emission and absorption techniques, electron beam-stimulated fluorescence, and light scattering - Rayleigh, Raman and Mie - methods. The following visualization methods are also discussed: shadow photography, schlieren photography, interferometry, holographic interferometry, laser anemometry, particle holography, and electron-excitation imaging. A large bibliography is presented and the work is copiously illustrated with figures and photographs