David M. Richards

The Ser/Thr protein kinase AfsK regulates polar growth and hyphal branching in the filamentous bacteria Streptomyces

In cells that exhibit apical growth, mechanisms that regulate cell polarity are crucial for determination of cellular shape and for the adaptation of growth to intrinsic and extrinsic cues. Broadly conserved pathways control cell polarity in eukaryotes, but less is known about polarly growing prokaryotes. An evolutionarily ancient form of apical growth is found in the filamentous bacteria Streptomyces, and is directed by a polarisome-like complex involving the essential protein DivIVA. We report here that this bacterial polarization machinery is regulated by a eukaryotic-type Ser/Thr protein kinase, AfsK, which localizes to hyphal tips and phosphorylates DivIVA. During normal growth, AfsK regulates hyphal branching by modulating branch-site selection and some aspect of the underlying polarisome-splitting mechanism that controls branching of Streptomyces hyphae. Further, AfsK is activated by signals generated by the arrest of cell wall synthesis and directly communicates this to the polarisome by hyperphosphorylating DivIVA. Induction of high levels of DivIVA phosphorylation by using a constitutively active mutant AfsK causes disassembly of apical polarisomes, followed by establishment of multiple hyphal branches elsewhere in the cell, revealing a profound impact of this kinase on growth polarity. The function of AfsK is reminiscent of the phoshorylation of polarity proteins and polarisome components by Ser/Thr protein kinases in eukaryotes.

Regulation of apical growth and hyphal branching in Streptomyces

The filamentous bacteria Streptomyces grow by tip extension and through the initiation of new branches, and this apical growth is directed by a polarisome-like complex involving the essential polarity protein DivIVA. New branch sites must be marked de novo and, until recently, there was no understanding of how these new sites are selected. Equally, hyphal branching patterns are affected by environmental conditions, but there was no insight into how polar growth and hyphal branching might be regulated in response to external or internal cues. This review focuses on recent discoveries that reveal the principal mechanism of branch site selection in Streptomyces, and the first mechanism to be identified that regulates polarisome behaviour to modulate polar growth and hyphal branching.

Mechanistic Basis of Branch-Site Selection in Filamentous Bacteria

Many filamentous organisms, such as fungi, grow by tip-extension and by forming new branches behind the tips. A similar growth mode occurs in filamentous bacteria, including the genus Streptomyces, although here our mechanistic understanding has been very limited. The Streptomyces protein DivIVA is a critical determinant of hyphal growth and localizes in foci at hyphal tips and sites of future branch development. However, how such foci form was previously unknown. Here, we show experimentally that DivIVA focus-formation involves a novel mechanism in which new DivIVA foci break off from existing tip-foci, bypassing the need for initial nucleation or de novo branch-site selection. We develop a mathematical model for DivIVA-dependent growth and branching, involving DivIVA focus-formation by tip-focus splitting, focus growth, and the initiation of new branches at a critical focus size. We quantitatively fit our model to the experimentally-measured tip-to-branch and branch-to-branch length distributions. The model predicts a particular bimodal tip-to-branch distribution results from tip-focus splitting, a prediction we confirm experimentally. Our work provides mechanistic understanding of a novel mode of hyphal growth regulation that may be widely employed.

The Mechanism of Phagocytosis: Two Stages of Engulfment

Despite being of vital importance to the immune system, the mechanism by which cells engulf relatively large solid particles during phagocytosis is still poorly understood. From movies of neutrophil phagocytosis of polystyrene beads, we measure the fractional engulfment as a function of time and demonstrate that phagocytosis occurs in two distinct stages. During the first stage, engulfment is relatively slow and progressively slows down as phagocytosis proceeds. However, at approximately half-engulfment, the rate of engulfment increases dramatically, with complete engulfment attained soon afterwards. By studying simple mathematical models of phagocytosis, we suggest that the first stage is due to a passive mechanism, determined by receptor diffusion and capture, whereas the second stage is more actively controlled, perhaps with receptors being driven toward the site of engulfment. We then consider a more advanced model that includes signaling and captures both stages of engulfment. This model predicts that there is an optimum ligand density for quick engulfment. Further, we show how this model explains why nonspherical particles engulf quickest when presented tip-first. Our findings suggest that active regulation may be a later evolutionary innovation, allowing fast and robust engulfment even for large particles.

Accurate encoding and decoding by single cells: amplitude versus frequency modulation.

Cells sense external concentrations and, via biochemical signaling, respond by regulating the expression of target proteins. Both in signaling networks and gene regulation there are two main mechanisms by which the concentration can be encoded internally: amplitude modulation (AM), where the absolute concentration of an internal signaling molecule encodes the stimulus, and frequency modulation (FM), where the period between successive bursts represents the stimulus. Although both mechanisms have been observed in biological systems, the question of when it is beneficial for cells to use either AM or FM is largely unanswered. Here, we first consider a simple model for a single receptor (or ion channel), which can either signal continuously whenever a ligand is bound, or produce a burst in signaling molecule upon receptor binding. We find that bursty signaling is more accurate than continuous signaling only for sufficiently fast dynamics. This suggests that modulation based on bursts may be more common in signaling networks than in gene regulation. We then extend our model to multiple receptors, where continuous and bursty signaling are equivalent to AM and FM respectively, finding that AM is always more accurate. This implies that the reason some cells use FM is related to factors other than accuracy, such as the ability to coordinate expression of multiple genes or to implement threshold crossing mechanisms.

Target shape dependence in a simple model of receptor-mediated endocytosis and phagocytosis

Significance Most previous work on endocytosis (the way cells ingest particles) has focused on engulfment of spherical targets. However, this is rarely relevant outside the laboratory, where cells must engulf a huge variety of target shapes, from capped cylinders (like many bacteria) to hourglasses (such as dividing cells). Here, we map this problem to the well-studied physics problem of the freezing of water. This allows us to study a wide range of different, biologically relevant shapes. In particular, for the first time, to our knowledge, it allows us to examine the orientation dependence of phagocytosis, to explain the shape dependence of phagocytosis and receptor-meditated endocytosis, and to explain why nonspherical targets often engulf faster when the most highly-curved tip is engulfed first. Phagocytosis and receptor-mediated endocytosis are vitally important particle uptake mechanisms in many cell types, ranging from single-cell organisms to immune cells. In both processes, engulfment by the cell depends critically on both particle shape and orientation. However, most previous theoretical work has focused only on spherical particles and hence disregards the wide-ranging particle shapes occurring in nature, such as those of bacteria. Here, by implementing a simple model in one and two dimensions, we compare and contrast receptor-mediated endocytosis and phagocytosis for a range of biologically relevant shapes, including spheres, ellipsoids, capped cylinders, and hourglasses. We find a whole range of different engulfment behaviors with some ellipsoids engulfing faster than spheres, and that phagocytosis is able to engulf a greater range of target shapes than other types of endocytosis. Further, the 2D model can explain why some nonspherical particles engulf fastest (not at all) when presented to the membrane tip-first (lying flat). Our work reveals how some bacteria may avoid being internalized simply because of their shape, and suggests shapes for optimal drug delivery.

Spatiotemporal analysis of different mechanisms for interpreting morphogen gradients.

During development, multicellular organisms must accurately control both temporal and spatial aspects of tissue patterning. This is often achieved using morphogens, signaling molecules that form spatially varying concentrations and so encode positional information. Typical analysis of morphogens assumes that spatial information is decoded in steady state by measuring the value of the morphogen concentration. However, recent experimental work suggests that both pre-steady-state readout and measurement of spatial and temporal derivatives of the morphogen concentration can play important roles in defining boundaries. Here, we undertake a detailed theoretical and numerical study of the accuracy of patterning-both in space and time-in models where readout is provided not by the morphogen concentration but by its spatial and temporal derivatives. In both cases we find that accurate patterning can be achieved, with sometimes even smaller errors than directly reading the morphogen concentration. We further demonstrate that such models provide other potential benefits to the system, such as the ability to switch on and off gene response with a high degree of spatiotemporal accuracy. Finally, we discuss how such derivatives might be calculated biologically and examine these models in relation to Sonic Hedgehog signaling in the vertebrate central nervous system. We show that, when coupled to a downstream transcriptional network, pre-steady-state measurement of the temporal change in the Shh morphogen is a plausible mechanism for determining precise gene boundaries in both space and time.

Quantitative Dynamics of Telomere Bouquet Formation

The mechanism by which homologous chromosomes pair during meiosis, as a prelude to recombination, has long been mysterious. At meiosis, the telomeres in many organisms attach to the nuclear envelope and move together to form the telomere bouquet, perhaps to facilitate the homologous search. It is believed that diffusion alone is not sufficient to account for the formation of the bouquet, and that some directed movement is also required. Here we consider the formation of the telomere bouquet in a wheat-rye hybrid both experimentally and using mathematical modelling. The large size of the wheat nucleus and wheats commercial importance make chromosomal pairing in wheat a particularly interesting and important process, which may well shed light on pairing in other organisms. We show that, prior to bouquet formation, sister chromatid telomeres are always attached to a hemisphere of the nuclear membrane and tend to associate in pairs. We study a mutant lacking the Ph1 locus, a locus ensuring correct homologous chromosome pairing, and discover that bouquet formation is delayed in the wild type compared to the mutant. Further, we develop a mathematical model of bouquet formation involving diffusion and directed movement, where we show that directed movement alone is sufficient to explain bouquet formation dynamics.

How cells engulf: a review of theoretical approaches to phagocytosis

Phagocytosis is a fascinating process whereby a cell surrounds and engulfs particles such as bacteria and dead cells. This is crucial both for single-cell organisms (as a way of acquiring nutrients) and as part of the immune system (to destroy foreign invaders). This whole process is hugely complex and involves multiple coordinated events such as membrane remodelling, receptor motion, cytoskeleton reorganisation and intracellular signalling. Because of this, phagocytosis is an excellent system for theoretical study, benefiting from biophysical approaches combined with mathematical modelling. Here, we review these theoretical approaches and discuss the recent mathematical and computational models, including models based on receptors, models focusing on the forces involved, and models employing energetic considerations. Along the way, we highlight a beautiful connection to the physics of phase transitions, consider the role of stochasticity, and examine links between phagocytosis and other types of endocytosis. We cover the recently discovered multistage nature of phagocytosis, showing that the size of the phagocytic cup grows in distinct stages, with an initial slow stage followed by a much quicker second stage starting around half engulfment. We also address the issue of target shape dependence, which is relevant to both pathogen infection and drug delivery, covering both one-dimensional and two-dimensional results. Throughout, we pay particular attention to recent experimental techniques that continue to inform the theoretical studies and provide a means to test model predictions. Finally, we discuss population models, connections to other biological processes, and how physics and modelling will continue to play a key role in future work in this area.

Illustrations of Mathematical Modeling in Biology: Epigenetics, Meiosis, and an Outlook

In the past few years, mathematical modeling approaches in biology have begun to fulfill their promise by assisting in the dissection of complex biological systems. Here, we review two recent examples of predictive mathematical modeling in plant biology. The first involves the quantitative epigenetic silencing of the floral repressor gene FLC in Arabidopsis, mediated by a Polycomb-based system. The second involves the spatiotemporal dynamics of telomere bouquet formation in wheat-rye meiosis. Although both the biology and the modeling framework of the two systems are different, both exemplify how mathematical modeling can help to accelerate discovery of the underlying mechanisms in complex biological systems. In both cases, the models that developed were relatively minimal, including only essential features, but both nevertheless yielded fundamental insights. We also briefly review the current state of mathematical modeling in biology, difficulties inherent in its application, and its potential future development.