Model for Protein Concentration Gradients in the Cytoplasm

Long-range formation of the Bicoid gradient requires multiple dynamic modes that spatially vary across the embryo

Morphogen gradients provide essential positional information to gene networks through their spatially heterogeneous distribution, yet how they form is still hotly contested, with multiple models proposed for different systems. Here, we focus on the transcription factor Bicoid (Bcd), a morphogen that forms an exponential gradient across the anterior-posterior (AP) axis of the early Drosophila embryo. Using fluorescence correlation spectroscopy we find there are spatial differences in Bcd diffusivity along the AP axis, with Bcd diffusing more rapidly in the posterior. We establish that such spatially varying differences in Bcd dynamics are sufficient to explain how Bcd can have a steep exponential gradient in the anterior half of the embryo and yet still have an observable fraction of Bcd near the posterior pole. In the nucleus, we demonstrate that Bcd dynamics are impacted by binding to DNA. Addition of the Bcd homeodomain to eGFP::NLS qualitatively replicates the Bcd concentration profile, suggesting this domain regulates Bcd dynamics. Our results reveal how a long-range gradient can form while retaining a steep profile through much of its range.

The architecture of redox microdomains: Cascading gradients and peroxiredoxins' redox-oligomeric coupling integrate redox signaling and antioxidant protection

In the cytosol of human cells under low oxidative loads, hydrogen peroxide is confined to microdomains around its supply sites, due to its fast consumption by peroxiredoxins. So are the sulfenic and disulfide forms of the 2-Cys peroxiredoxins, according to a previous theoretical analysis [Travasso et al., Redox Biology 15 (2017) 297]. Here, an extended reaction-diffusion model that for the first time considers the differential properties of human peroxiredoxins 1 and 2 and the thioredoxin redox cycle predicts important new aspects of the dynamics of redox microdomains. The peroxiredoxin 1 sulfenates and disulfides are more localized than the corresponding peroxiredoxin 2 forms, due to the former peroxiredoxin's faster resolution step. The thioredoxin disulfides are also localized. As the H2O2 supply rate (vsup) approaches and then surpasses the maximal rate of the thioredoxin/thioredoxin reductase system (V), these concentration gradients become shallower, and then vanish. At low vsup the peroxiredoxin concentration determines the H2O2 concentrations and gradient length scale, but as vsup approaches V, the thioredoxin reductase activity gains influence. A differential mobility of peroxiredoxin disulfide dimers vs. reduced decamers enhances the redox polarity of the cytosol: as vsup approaches V, reduced decamers are preferentially retained far from H2O2 sources, attenuating the local H2O2 buildup. Substantial total protein concentration gradients of both peroxiredoxins emerge under these conditions, and the concentration of reduced peroxiredoxin 1 far from the H2O2 sources even increases with vsup. Altogether, the properties of 2-Cys peroxiredoxins and thioredoxin are such that localized H2O2 supply induces a redox and functional polarization between source-proximal regions (redox microdomains) that facilitate peroxiredoxin-mediated signaling and distal regions that maximize antioxidant protection.

Order from chaos: cellular asymmetries explained with modelling

Molecules inside cells are subject to physical forces and undergo biochemical interactions, continuously changing their physical properties and dynamics. Despite this, cells achieve highly ordered molecular patterns that are crucial to regulate various cellular functions and to specify cell fate. In the Caenorhabditis elegans one-cell embryo, protein asymmetries are established in the narrow time window of a cell division. What are the mechanisms that allow molecules to establish asymmetries, defying the randomness imposed by Brownian motion? Mathematical and computational models have paved the way to the understanding of protein dynamics up to the 'single-molecule level' when resolution represents an issue for precise experimental measurements. Here we review the models that interpret cortical and cytoplasmic asymmetries in the one-cell C. elegans embryo.

Is there a localized role for translational quality control?

It is known that mRNAs and the machinery that translates them are not uniformly distributed throughout the cytoplasm. As a result, the expression of some genes is localized to particular parts of the cell and this makes it possible to carry out important activities, such as growth and signaling, in three-dimensional space. However, the functions of localized gene expression are not fully understood, and the underlying mechanisms that enable localized expression have not been determined in many cases. One consideration that could help in addressing these challenges is the role of quality control (QC) mechanisms that monitor translating ribosomes. On a global level, QC pathways are critical for detecting aberrant translation events, such as a ribosome that stalls while translating, and responding by activating stress pathways and resolving problematic ribosomes and mRNAs at the molecular level. However, it is unclear how these pathways, even when uniformly active throughout the cell, affect local translation. Importantly, some QC pathways have themselves been reported to be enriched in the proximity of particular organelles, but the extent of such localized activity remains largely unknown. Here, we describe the major QC pathways and review studies that have begun to explore their roles in localized translation. Given the limited data in this area, we also pose broad questions about the possibilities and limitations for how QC pathways could facilitate localized gene expression in the cell with the goal of offering ideas for future experimentation.

Modeling protein dynamics in Caenorhabditis elegans embryos reveals that the PLK-1 gradient relies on weakly coupled reaction-diffusion mechanisms

Intracellular gradients have essential roles in cell and developmental biology, but their formation is not fully understood. We have developed a computational approach facilitating interpretation of protein dynamics and gradient formation. We have combined this computational approach with experiments to understand how Polo-Like Kinase 1 (PLK-1) forms a cytoplasmic gradient in Caenorhabditis elegans embryos. Although the PLK-1 gradient depends on the Muscle EXcess-5/6 (MEX-5/6) proteins, we reveal differences in PLK-1 and MEX-5 gradient formation that can be explained by a model with two components, PLK-1 bound to MEX-5 and unbound PLK-1. Our combined approach suggests that a weak coupling between PLK-1 and MEX-5 reaction–diffusion mechanisms dictates the dynamic exchange of PLK-1 with the cytoplasm, explaining PLK-1 high diffusivity and smooth gradient.

Protein gradients have fundamental roles in cell and developmental biology. In the one-cell Caenorhabditis elegans embryo, the mitotic Polo-Like Kinase 1 (PLK-1) forms an anterior-rich cytoplasmic gradient, which is crucial for asymmetric cell division and embryonic development. The PLK-1 gradient depends on the RNA-binding Muscle-EXcess-5 protein (MEX-5), whose slow-diffusing complexes accumulate in the anterior via a reaction–diffusion mechanism. Here, we combine experiments and a computational approach to investigate the dynamics of PLK-1 gradient formation. We find that the gradient of PLK-1 initiates later, is less steep, and forms with slower dynamics than does the MEX-5 gradient. The data show that PLK-1 diffuses faster than MEX-5 in both anterior and posterior cytoplasmic regions. Our simulations suggest that binding to slow-diffusing MEX-5 is required for PLK-1 gradient formation, but that a significant fraction of unbound PLK-1 is necessary to justify the different gradient dynamics. We provide a computational tool able to predict gradient establishment prior to cell division and show that a two-component, bound and unbound, model of PLK-1 dynamics recapitulates the experimental observations.

Anterior-posterior pattern formation in ciliates

As single cells, ciliates build, duplicate, and even regenerate complex cortical patterns by largely unknown mechanisms that precisely position organelles along two cell‐wide axes: anterior–posterior and circumferential (left–right). We review our current understanding of intracellular patterning along the anterior–posterior axis in ciliates, with emphasis on how the new pattern emerges during cell division. We focus on the recent progress at the molecular level that has been driven by the discovery of genes whose mutations cause organelle positioning defects in the model ciliate Tetrahymena thermophila. These investigations have revealed a network of highly conserved kinases that are confined to either anterior or posterior domains in the cell cortex. These pattern‐regulating kinases create zones of cortical inhibition that by exclusion determine the precise placement of organelles. We discuss observations and models derived from classical microsurgical experiments in large ciliates (including Stentor) and interpret them in light of recent molecular findings in Tetrahymena. In particular, we address the involvement of intracellular gradients as vehicles for positioning organelles along the anterior‐posterior axis.

Mechanisms of Viscous Media Effects on Elementary Steps of Bacterial Bioluminescent Reaction

Enzymes activity in a cell is determined by many factors, among which viscosity of the microenvironment plays a significant role. Various cosolvents can imitate intracellular conditions in vitro, allowing to reduce a combination of different regulatory effects. The aim of the study was to analyze the media viscosity effects on the rate constants of the separate stages of the bacterial bioluminescent reaction. Non-steady-state reaction kinetics in glycerol and sucrose solutions was measured by stopped-flow technique and analyzed with a mathematical model developed in accordance with the sequence of reaction stages. Molecular dynamics methods were applied to reveal the effects of cosolvents on luciferase structure. We observed both in glycerol and in sucrose media that the stages of luciferase binding with flavin and aldehyde, in contrast to oxygen, are diffusion-limited. Moreover, unlike glycerol, sucrose solutions enhanced the rate of an electronically excited intermediate formation. The MD simulations showed that, in comparison with sucrose, glycerol molecules could penetrate the active-site gorge, but sucrose solutions caused a conformational change of functionally important αGlu175 of luciferase. Therefore, both cosolvents induce diffusion limitation of substrates binding. However, in sucrose media, increasing enzyme catalytic constant neutralizes viscosity effects. The activating effect of sucrose can be attributed to its exclusion from the catalytic gorge of luciferase and promotion of the formation of the active site structure favorable for the catalysis.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Persistent growth of microtubules at low density

Microtubules (MTs) often form a polarized array with minus ends anchored at the centrosome and plus ends extended toward the cell margins. Plus ends display behavior known as dynamic instability—transitions between rapid shortening and slow growth. It is known that dynamic instability is regulated locally to ensure entry of MTs into nascent areas of the cytoplasm, but details of this regulation remain largely unknown. Here, we test an alternative hypothesis for the local regulation of MT behavior. We used microsurgery to isolate a portion of peripheral cytoplasm from MTs growing from the centrosome, creating cytoplasmic areas locally depleted of MTs. We found that in sparsely populated areas MT plus ends persistently grew or paused but never shortened. In contrast, plus ends that entered regions of cytoplasm densely populated with MTs frequently transitioned to shortening. Persistent growth of MTs in sparsely populated areas could not be explained by a local increase in concentration of free tubulin subunits or elevation of Rac1 activity proposed to enhance MT growth at the cell leading edge during locomotion. These observations suggest the existence of a MT density–dependent mechanism regulating MT dynamics that determines dynamic instability of MTs in densely populated areas of the cytoplasm and persistent growth in sparsely populated areas.

A Disorder-to-Order Transition Mediates RNA Binding of the Caenorhabditis elegans Protein MEX-5

CCCH-type tandem zinc finger (TZF) domains are found in many RNA-binding proteins (RBPs) that regulate the essential processes of post-transcriptional gene expression and splicing through direct protein-RNA interactions. In Caenorhabditis elegans, RBPs control the translation, stability, or localization of maternal messenger RNAs required for patterning decisions before zygotic gene activation. MEX-5 (Muscle EXcess) is a C. elegans protein that leads a cascade of RBP localization events that is essential for axis polarization and germline differentiation after fertilization. Here, we report that at room temperature, the CCCH-type TZF domain of MEX-5 contains an unstructured zinc finger that folds upon binding of its RNA target. We have characterized the structure and dynamics of the TZF domain of MEX-5 and designed a variant MEX-5 in which both fingers are fully folded in the absence of RNA. Within the thermal range experienced by C. elegans, the population of the unfolded state of the TZF domain of MEX-5 varies. We observe that the TZF domain becomes less disordered at lower temperatures and more disordered at higher temperatures. However, in the temperature range in which C. elegans is fertile, when MEX-5 needs to be functional, only one of the two zinc fingers is folded.

Intracellular morphogens: Specifying patterns at the subcellular scale

The notion that graded distributions of signals underlie the spatial organization of biological systems has long been a central pillar in the fields of cell and developmental biology. During morphogenesis, morphogens spread across tissues to guide development of the embryo. Similarly, a variety of dynamic gradients and pattern-forming networks have been discovered that shape subcellular organization. Here we discuss the principles of intracellular pattern formation by these intracellular morphogens and relate them to conceptually similar processes operating at the tissue scale. We will specifically review mechanisms for generating cellular asymmetry and consider how intracellular patterning networks are controlled and adapt to cellular geometry. Finally, we assess the general concept of intracellular gradients as a mechanism for positional control in light of current data, highlighting how the simple readout of fixed concentration thresholds fails to fully capture the complexity of spatial patterning processes occurring inside cells.

Spatial regulation of the polarity kinase PAR-1 by parallel inhibitory mechanisms

The MARK/PAR-1 family of kinases are conserved regulators of cell polarity that share a conserved C-terminal kinase-associated domain (KA1). Localization of MARK/PAR-1 kinases to specific regions of the cell cortex is a hallmark of polarized cells. In Caenorhabditiselegans zygotes, PAR-1 localizes to the posterior cortex under the influence of another polarity kinase, aPKC/PKC-3. Here, we report that asymmetric localization of PAR-1 protein is not essential, and that PAR-1 kinase activity is regulated spatially. We find that, as in human MARK1, the PAR-1 KA1 domain is an auto-inhibitory domain that suppresses kinase activity. Auto-inhibition by the KA1 domain functions in parallel with phosphorylation by PKC-3 to suppress PAR-1 activity in the anterior cytoplasm. The KA1 domain also plays an additional role that is essential for germ plasm maintenance and fertility. Our findings suggest that modular regulation of kinase activity by redundant inhibitory inputs contributes to robust symmetry breaking by MARK/PAR-1 kinases in diverse cell types.

How Important Is Protein Diffusion in Prokaryotes?

That diffusion is important for the proper functioning of cells is without question. The extent to which the diffusion coefficient is important is explored here for prokaryotic cells. We discuss the principles of diffusion focusing on diffusion-limited reactions, summarize the known values for diffusion coefficients in prokaryotes and in in vitro model systems, and explain a number of cases where diffusion coefficients are either limiting for reaction rates or necessary for the existence of phenomena. We suggest a number of areas that need further study including expanding the range of organism growth temperatures, direct measurements of diffusion limitation, expanding the range of cell sizes, diffusion limitation for membrane proteins, and taking into account cellular context when assessing the possibility of diffusion limitation.

Rapid diffusion-state switching underlies stable cytoplasmic gradients in the Caenorhabditis elegans zygote

Protein concentration gradients organize cells and tissues and commonly form through diffusion away from a local source of protein. Interestingly, during the asymmetric division of the Caenorhabditis elegans zygote, the RNA-binding proteins MEX-5 and PIE-1 form opposing concentration gradients in the absence of a local source. In this study, we use near-total internal reflection fluorescence (TIRF) imaging and single-particle tracking to characterize the reaction/diffusion dynamics that maintain the MEX-5 and PIE-1 gradients. Our findings suggest that both proteins interconvert between fast-diffusing and slow-diffusing states on timescales that are much shorter (seconds) than the timescale of gradient formation (minutes). The kinetics of diffusion-state switching are strongly polarized along the anterior/posterior (A/P) axis by the PAR polarity system such that fast-diffusing MEX-5 and PIE-1 particles are approximately symmetrically distributed, whereas slow-diffusing particles are highly enriched in the anterior and posterior cytoplasm, respectively. Using mathematical modeling, we show that local differences in the kinetics of diffusion-state switching can rapidly generate stable concentration gradients over a broad range of spatial and temporal scales.

Modeling the influence of co-localized intracellular calcium stores on the secretory response of bovine chromaffin cells

Catecholamines secretion from chromaffin cells is mediated by a Ca²⁺-dependent process in the submembrane space where the exocytotic machinery is located and high-Ca²⁺ microdomains (HCMDs) are formed by the coordinated activity of a functional triad composed of Ca²⁺ channels, endoplasmic reticulum (ER) and mitochondria. It has been observed experimentally that subpopulations of cortical mitochondria and ER associate to secretory sites in bovine chromaffin cells. Here, we study the effect of the geometrical distribution of the co-localized cortical organelles both in the formation of HCMDs in the vicinity of Ca²⁺ channels and on the secretory activity of bovine chromaffin cells in response to a single voltage pulse. Our simulations indicate that co-localized organelles have a dual role in the formation of HCMDs, having, on the one hand, an amplification effect due to the Ca²⁺-induced Ca²⁺-release mechanism from the ER and, on the other, acting as physical barriers to Ca²⁺ diffusion. In addition, our simulations suggest that the increased levels of Ca²⁺ in the microdomain enhances the secretion of the vesicles co-localized to the Ca²⁺ channels. As a whole, our results support the idea that the functional triads formed by Ca²⁺ channels, subplasmalemma ER and mitochondria have a positive effect on the secretion of catecholamines in bovine chromaffin cells.

Protein gradients in single cells induced by their coupling to "morphogen"-like diffusion

One of the many ways cells transmit information within their volume is through steady spatial gradients of different proteins. However, the mechanism through which proteins without any sources or sinks form such single-cell gradients is not yet fully understood. One of the models for such gradient formation, based on differential diffusion, is limited to proteins with large ratios of their diffusion constants or to specific protein-large molecule interactions. We introduce a novel mechanism for gradient formation via the coupling of the proteins within a single cell with a molecule, that we call a "pronogen," whose action is similar to that of morphogens in multi-cell assemblies; the pronogen is produced with a fixed flux at one side of the cell. This coupling results in an effectively non-linear diffusion degradation model for the pronogen dynamics within the cell, which leads to a steady-state gradient of the protein concentration. We use stability analysis to show that these gradients are linearly stable with respect to perturbations.

Polo-like Kinase Couples Cytoplasmic Protein Gradients in the C. elegans Zygote

Intracellular protein gradients underlie essential cellular and developmental processes, but the mechanisms by which they are established are incompletely understood. During the asymmetric division of the C. elegans zygote, the RNA-binding protein MEX-5 forms an anterior-rich cytoplasmic gradient that causes the RNA-binding protein POS-1 to form an opposing, posterior-rich gradient. We demonstrate that the polo-like kinase PLK-1 mediates the repulsive coupling between MEX-5 and POS-1 by increasing the mobility of POS-1 in the anterior. PLK-1 is enriched in the anterior cytoplasm and phosphorylates POS-1, which is both necessary and sufficient to increase POS-1 mobility. Regulation of POS-1 mobility depends on both the interaction between PLK-1 and MEX-5 and between MEX-5 and RNA, suggesting that MEX-5 may recruit PLK-1 to RNA in the anterior. The low concentration of MEX-5/PLK-1 in the posterior cytoplasm provides a permissive environment for the retention of POS-1, which depends on POS-1 RNA binding. Our findings describe a novel reaction/diffusion mechanism in which the asymmetric distribution of cytoplasmic PLK-1 couples two RNA-binding protein gradients, thereby partitioning the cytoplasm.

Regulation of Cell Polarity by PAR-1/MARK Kinase

PAR-1/MARK kinases are conserved serine/threonine kinases that are essential regulators of cell polarity. PAR-1/MARK kinases localize and function in opposition to the anterior PAR proteins to control the asymmetric distribution of factors in a wide variety polarized cells. In this review, we discuss the mechanisms that control the localization and activity of PAR-1/MARK kinases, including their antagonistic interactions with the anterior PAR proteins. We focus on the role PAR-1 plays in the asymmetric division of the Caenorhabditis elegans zygote, in the establishment of the anterior/posterior axis in the Drosophila oocyte and in the control of microtubule dynamics in mammalian neurons. In addition to conserved aspects of PAR-1 biology, we highlight the unique ways in which PAR-1 acts in these distinct cell types to orchestrate their polarization. Finally, we review the connections between disruptions in PAR-1/MARK function and Alzheimer's disease and cancer.

Model of Growth Cone Membrane Polarization via Microtubule Length Regulation

We present a mathematical model of membrane polarization in growth cones. We proceed by coupling an active transport model of cytosolic proteins along a two-dimensional microtubule (MT) network with a modified Dogterom-Leibler model of MT growth. In particular, we consider a Rac1-stathmin-MT pathway in which the growth and catastrophe rates of MTs are regulated by cytosolic stathmin, while the stathmin is regulated by Rac1 at the membrane. We use regular perturbation theory and numerical simulations to determine the steady-state stathmin concentration, the mean MT length distribution, and the resulting distribution of membrane-bound proteins. We thus show how a nonuniform Rac1 distribution on the membrane generates a polarized distribution of membrane proteins. The mean MT length distribution and hence the degree of membrane polarization are sensitive to the precise form of the Rac1 distribution and parameters such as the catastrophe-promoting constant and tubulin association rate. This is a consequence of the fact that the lateral diffusion of stathmin tends to weaken the effects of Rac1 on the distribution of mean MT lengths.

Spatial organization of the Ran pathway by microtubules in mitosis

Concentration gradients of soluble proteins are believed to be responsible for control of morphogenesis of subcellular systems, but the mechanisms that generate the spatial organization of these subcellular gradients remain poorly understood. Here, we use a newly developed multipoint fluorescence fluctuation spectroscopy technique to study the ras-related nuclear protein (Ran) pathway, which forms soluble gradients around chromosomes in mitosis and is thought to spatially regulate microtubule behaviors during spindle assembly. We found that the distribution of components of the Ran pathway that influence microtubule behaviors is determined by their interactions with microtubules, resulting in microtubule nucleators being localized by the microtubules whose formation they stimulate. Modeling and perturbation experiments show that this feedback makes the length of the spindle insensitive to the length scale of the Ran gradient, allows the spindle to assemble outside the peak of the Ran gradient, and explains the scaling of the spindle with cell size. Such feedback between soluble signaling pathways and the mechanics of the cytoskeleton may be a general feature of subcellular organization.

Dynamics of gradient formation by intracellular shuttling

A number of important cellular functions rely on the formation of intracellular protein concentration gradients. Experimental studies discovered a number of mechanisms for the formation of such gradients. One of the mechanisms relies on the intracellular shuttling of a protein that interconverts between the two states with different diffusivities, under the action of two enzymes, one of which is localized to the plasma membrane, whereas the second is uniformly distributed in the cytoplasm. Recent work reported an analytical solution for the steady state gradient in this mechanism, obtained in the framework of a one-dimensional reaction-diffusion model. Here, we study the dynamics in this model and derive analytical expressions for the Laplace transforms of the time-dependent concentration profiles in terms of elementary transcendental functions. Inverting these transforms numerically, one can obtain time-dependent concentration profiles of the two forms of the protein.

Coupling between cytoplasmic concentration gradients through local control of protein mobility in the Caenorhabditis elegans zygote

Cell polarity is characterized by the asymmetric distribution of factors at the cell cortex and in the cytoplasm. Although mechanisms that establish cortical asymmetries have been characterized, less is known about how persistent cytoplasmic asymmetries are generated. During the asymmetric division of the Caenorhabditis elegans zygote, the PAR proteins orchestrate the segregation of the cytoplasmic RNA-binding proteins MEX-5/6 to the anterior cytoplasm and PIE-1, POS-1, and MEX-1 to the posterior cytoplasm. In this study, we find that MEX-5/6 control the segregation of GFP::PIE-1, GFP::POS-1, and GFP::MEX-1 by locally increasing their mobility in the anterior cytoplasm. Remarkably, PIE-1, POS-1, and MEX-1 form gradients with distinct strengths, which correlates with differences in their responsiveness to MEX-5/6. We show that MEX-5/6 act downstream of the polarity regulators PAR-1 and PAR-3 and in a concentration-dependent manner to increase the mobility of GFP::PIE-1. These findings suggest that the MEX-5/6 concentration gradients are directly coupled to the establishment of posterior-rich PIE-1, POS-1, and MEX-1 concentration gradients via the formation of anterior-fast, posterior-slow mobility gradients.

Cytoplasmic localization and asymmetric division in the early embryo of Caenorhabditis elegans

During the initial cleavages of the Caenorhabditis elegans embryo, a series of rapid and invariant asymmetric cell divisions pattern the fate, size, and position of four somatic blastomeres and a single germline blastomere. These asymmetric divisions are orchestrated by a collection of maternally deposited factors that are initially symmetrically distributed in the newly fertilized embryo. Maturation of the sperm-derived centrosome in the posterior cytoplasm breaks this symmetry by triggering a dramatic and highly stereotyped partitioning of these maternal factors. A network of conserved cell polarity regulators, the PAR proteins, form distinct anterior and posterior domains at the cell cortex. From these domains, the PAR proteins direct the segregation of somatic and germline factors into opposing regions of the cytoplasm such that, upon cell division, they are preferentially inherited by the somatic blastomere or the germline blastomere, respectively. The segregation of these factors is controlled, at least in part, by a series of reaction-diffusion mechanisms that are asymmetrically deployed along the anterior/posterior axis. The characterization of these mechanisms has important implications for our understanding of how cells are polarized and how spatial organization is generated in the cytoplasm. For further resources related to this article, please visit the WIREs website.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

cellPACK: a virtual mesoscope to model and visualize structural systems biology

cellPACK assembles computational models of the biological mesoscale, an intermediate scale (10⁻⁷–10⁻⁸m) between molecular and cellular biology. cellPACK’s modular architecture unites existing and novel packing algorithms to generate, visualize and analyze comprehensive 3D models of complex biological environments that integrate data from multiple experimental systems biology and structural biology sources. cellPACK is currently available as open source code, with tools for validation of models and with recipes and models for five biological systems: blood plasma, cytoplasm, synaptic vesicles, HIV and a mycoplasma cell. We have applied cellPACK to model distributions of HIV envelope protein to test several hypotheses for consistency with experimental observations. Biologists, educators, and outreach specialists can interact with cellPACK models, develop new recipes and perform packing experiments through scripting and graphical user interfaces at http://cellPACK.org.

Covalent modification cycles through the spatial prism

Covalent modification cycles are basic units and building blocks of posttranslational modification and cellular signal transduction. We systematically explore different spatial aspects of signal transduction in covalent modification cycles by starting with a basic temporal cycle as a reference and focusing on steady-state signal transduction. We consider, in turn, the effect of diffusion on spatial signal transduction, spatial analogs of ultrasensitive behavior, and the interplay between enzyme localization and substrate diffusion. Our analysis reveals the need to explicitly account for kinetics and diffusional transport (and localization) of enzymes, substrates, and complexes. It demonstrates a complex and subtle interplay between spatial heterogeneity, diffusion, and localization. Overall, examining the spatial dimension of covalent modification reveals that 1), there are important differences between spatial and temporal signal transduction even in this cycle; and 2), spatial aspects may play a substantial role in affecting and distorting information transfer in modules/networks that are usually studied in purely temporal terms. This has important implications for the systematic understanding of signaling in covalent modification cycles, pathways, and networks in multiple cellular contexts.

KNL1 facilitates phosphorylation of outer kinetochore proteins by promoting Aurora B kinase activity

KNL1 is essential for efficient kinetochore–microtubule attachment dynamics during mitosis, in part through promotion of Bub1-mediated targeting of Aurora B to the kinetochore.

Aurora B kinase phosphorylates kinetochore proteins during early mitosis, increasing kinetochore–microtubule (MT) turnover and preventing premature stabilization of kinetochore–MT attachments. Phosphorylation of kinetochore proteins during late mitosis is low, promoting attachment stabilization, which is required for anaphase onset. The kinetochore protein KNL1 recruits Aurora B–counteracting phosphatases and the Aurora B–targeting factor Bub1, yet the consequences of KNL1 depletion on Aurora B phospho-regulation remain unknown. Here, we demonstrate that the KNL1 N terminus is essential for Aurora B activity at kinetochores. This region of KNL1 is also required for Bub1 kinase activity at kinetochores, suggesting that KNL1 promotes Aurora B activity through Bub1-mediated Aurora B targeting. However, ectopic targeting of Aurora B to kinetochores does not fully rescue Aurora B activity in KNL1-depleted cells, suggesting KNL1 influences Aurora B activity through an additional pathway. Our findings establish KNL1 as a requirement for Aurora B activity at kinetochores and for wild-type kinetochore–MT attachment dynamics.

Smad2 and Smad3 cooperate and antagonize simultaneously in vertebrate neurogenesis

The transforming growth factor beta (TGF-β) pathway plays key roles in development and cancer. (TGF-β) signaling converges on the Smad2 and Smad3 effectors, which can either cooperate or antagonize to regulate their transcriptional targets. Here we performed in vivo and in silico experiments to study how such cooperativity and antagonism might function during neurogenesis. In vivo electroporation experiments in the chick embryo neural tube show that Smad2 and Smad3 cooperate to promote neurogenesis, as well as the transcription of Smad3 specific targets. Smad2 knockdown enhances neurogenesis and the transcription of Smad3 specific targets. A mathematical model of the TGF-β pathway fits the experimental results and predicts that the proportions of the three different trimeric complexes formed dictates the transcriptional responses of the R-Smads. As such, Smad2 targets are activated solely by the Smad2-Smad2-Smad4 complex, while Smad3 targets are activated both by Smad2-Smad3 Smad4 and Smad3- Smad3-Smad4 trimers. Since we have modeled the Smad responses onto arbitrary genes, we propose that this mechanism might be extended to additional activities of TGF-β in development and disease.

Physical constraints on the establishment of intracellular spatial gradients in bacteria

Background

Bacteria dynamically regulate their intricate intracellular organization involving proteins that facilitate cell division, motility, and numerous other processes. Consistent with this sophisticated organization, bacteria are able to create asymmetries and spatial gradients of proteins by localizing signaling pathway components. We use mathematical modeling to investigate the biochemical and physical constraints on the generation of intracellular gradients by the asymmetric localization of a source and a sink.

Results

We present a systematic computational analysis of the effects of other regulatory mechanisms, such as synthesis, degradation, saturation, and cell growth. We also demonstrate that gradients can be established in a variety of bacterial morphologies such as rods, crescents, spheres, branched and constricted cells.

Conclusions

Taken together, these results suggest that gradients are a robust and potentially common mechanism for providing intracellular spatial cues.

Smoldyn on graphics processing units: massively parallel Brownian dynamics simulations

Space is a very important aspect in the simulation of biochemical systems; recently, the need for simulation algorithms able to cope with space is becoming more and more compelling. Complex and detailed models of biochemical systems need to deal with the movement of single molecules and particles, taking into consideration localized fluctuations, transportation phenomena, and diffusion. A common drawback of spatial models lies in their complexity: models can become very large, and their simulation could be time consuming, especially if we want to capture the systems behavior in a reliable way using stochastic methods in conjunction with a high spatial resolution. In order to deliver the promise done by systems biology to be able to understand a system as whole, we need to scale up the size of models we are able to simulate, moving from sequential to parallel simulation algorithms. In this paper, we analyze Smoldyn, a widely diffused algorithm for stochastic simulation of chemical reactions with spatial resolution and single molecule detail, and we propose an alternative, innovative implementation that exploits the parallelism of Graphics Processing Units (GPUs). The implementation executes the most computational demanding steps (computation of diffusion, unimolecular, and bimolecular reaction, as well as the most common cases of molecule-surface interaction) on the GPU, computing them in parallel on each molecule of the system. The implementation offers good speed-ups and real time, high quality graphics output

How to build a robust intracellular concentration gradient

Concentration gradients of morphogens are critical regulators of patterning in developmental biology. Increasingly, intracellular concentration gradients have also been found to orchestrate spatial organization, but inside single cells, where they regulate processes such as cell division, polarity and mitotic spindle dynamics. Here, we discuss recent progress in understanding how such intracellular gradients can be built robustly. We focus particularly on the Pom1p gradient in fission yeast, elucidating how various buffering mechanisms operate to ensure precise gradient formation. In this case, a systems-level understanding of the entire mechanism of precise gradient construction is now within reach, with important implications for gradients in both intracellular and developmental contexts.

Spatial and stochastic cellular modeling with the Smoldyn simulator

This chapter describes how to use Smoldyn, which is a computer program for modeling cellular systems with spatial and stochastic detail. Smoldyn represents each molecule of interest as an individual point-like particle. These simulated molecules diffuse, interact with surfaces (e.g., biological membranes), and undergo chemical reactions much as they would in real biochemical systems. Smoldyn has been used to model signal transduction within bacterial cells, pheromone signaling between yeast cells, bacterial carboxysome function, diffusion in crowded spaces, and many other systems. A new "rule-based modeling" feature automatically generates chemical species and reactions as they arise in simulations due to protein modifications and complexation. Smoldyn is easy to use, quantitatively accurate, and computationally efficient. It is generally best for systems with length scales between nanometers and several microns, time scales from tens of nanoseconds to tens of minutes, and up to about 10(5) individual molecules. Smoldyn runs on Macintosh, Linux, or Windows systems, is open source, and can be downloaded from http://www.smoldyn.org.

Regulation of the MEX-5 gradient by a spatially segregated kinase/phosphatase cycle

Protein concentration gradients encode spatial information across cells and tissues and often depend on spatially localized protein synthesis. Here, we report that a different mechanism underlies the MEX-5 gradient. MEX-5 is an RNA-binding protein that becomes distributed in a cytoplasmic gradient along the anterior-to-posterior axis of the one-cell C. elegans embryo. We demonstrate that the MEX-5 gradient is a direct consequence of an underlying gradient in MEX-5 diffusivity. The MEX-5 diffusion gradient arises when the PAR-1 kinase stimulates the release of MEX-5 from slow-diffusive, RNA-containing complexes in the posterior cytoplasm. PAR-1 directly phosphorylates MEX-5 and is antagonized by the spatially uniform phosphatase PP2A. Mathematical modeling and in vivo observations demonstrate that spatially segregated phosphorylation and dephosphorylation reactions are sufficient to generate stable protein concentration gradients in the cytoplasm. The principles demonstrated here apply to any spatially segregated modification cycle that affects protein diffusion and do not require protein synthesis or degradation.

Precision of sensing cell length via concentration gradients

Unicellular organisms are typically found to have a characteristic cell size. To achieve a homeostatic distribution of cell sizes over many generations requires that cell length is actively sensed and regulated. However, the mechanisms by which cell size is controlled remain poorly understood. Recent experiments in fission yeast have shown that cell length is controlled in part by polar gradients of the protein Pom1 together with localized measurement of concentration at midcell. Dilution as the cell grows leads to a reduction in the midcell protein concentration, which lifts a block on mitosis. Here we analyze the precision of this mechanism for length sensing in the presence of inevitable intrinsic noise in the processes leading to formation and measurement of this gradient. We find that the use of concentration gradients allows for more robust length sensing than a comparable spatially uniform system, and allows for reliable length determination even if the average protein concentration throughout the cell remains constant as the cell grows. Optimal values for the gradient decay length and receptor dissociation constant emerge from maximizing sensitivity while minimizing the impact of density fluctuations.

MEX-5 enrichment in the C. elegans early embryo mediated by differential diffusion

Specification of germline and somatic cell lineages in C. elegans originates in the polarized single-cell zygote. Several cell-fate determinants are partitioned unequally along the anterior-posterior axis of the zygote, ensuring the daughter cells a unique inheritance upon asymmetric cell division. Recent studies have revealed that partitioning of the germline determinant PIE-1 and the somatic determinant MEX-5 involve protein redistribution accompanied by spatiotemporal changes in protein diffusion rates. Here, we characterize the dynamics of MEX-5 in the zygote and propose a novel reaction/diffusion model to explain both its anterior enrichment and its remarkable intracellular dynamics without requiring asymmetrically distributed binding sites. We propose that asymmetric cortically localized PAR proteins mediate the anterior enrichment of MEX-5 by reversibly changing its diffusion rate at spatially distinct points in the embryo, thus generating a stable concentration gradient along the anterior-posterior axis of the cell. This work extends the scope of reaction/diffusion models to include not only germline morphogens, but also somatic determinants.

Detailed simulations of cell biology with Smoldyn 2.1

Most cellular processes depend on intracellular locations and random collisions of individual protein molecules. To model these processes, we developed algorithms to simulate the diffusion, membrane interactions, and reactions of individual molecules, and implemented these in the Smoldyn program. Compared to the popular MCell and ChemCell simulators, we found that Smoldyn was in many cases more accurate, more computationally efficient, and easier to use. Using Smoldyn, we modeled pheromone response system signaling among yeast cells of opposite mating type. This model showed that secreted Bar1 protease might help a cell identify the fittest mating partner by sharpening the pheromone concentration gradient. This model involved about 200,000 protein molecules, about 7000 cubic microns of volume, and about 75 minutes of simulated time; it took about 10 hours to run. Over the next several years, as faster computers become available, Smoldyn will allow researchers to model and explore systems the size of entire bacterial and smaller eukaryotic cells.

Accurate particle-based simulation of adsorption, desorption and partial transmission

Particle-based simulators represent molecules of interest with point-like particles that diffuse and react in continuous space. These simulators are often used to investigate spatial or stochastic aspects of biochemical systems. This paper presents new particle-based simulation algorithms for modeling interactions between molecules and surfaces; they address irreversible and reversible molecular adsorption to, desorption from and transmission through membranes. Their central elements are: (i) relationships between adsorption, desorption and transmission coefficients on the one hand, and simulator interaction probabilities on the other, and (ii) probability densities for initial placements of desorbed molecules. These algorithms, which were implemented and tested in the Smoldyn simulator, are accurate, easy to implement and computationally efficient. They allow longer time steps and better address reversible processes than an algorithm that Erban and Chapman recently presented (Physical Biology 4:16-28, 2007). This paper also presents a method for simulating unbounded diffusion in a limited spatial domain using a partially absorbing boundary, as well as new solutions to the diffusion differential equation with reversible Robin boundary conditions.

Introducing simulated cellular architecture to the quantitative analysis of fluorescent microscopy

Biological cells are complex and highly dynamic: many macromolecules are organized in loose assemblies, clusters or highly structured complexes, others exist most of the time as freely diffusing monomers. They move between regions and compartments through diffusion and enzyme-mediated transport, within a heavily crowded cytoplasm. To make sense of this complexity, computational models, and, in turn, quantitative in vivo data are needed. An array of fluorescent microscopy methods is available, but due to the inherent noise and complexity inside the cell, they are often hard to interpret. Using the example of fluorescence recovery after photobleaching (FRAP) and the bacterial chemotaxis system, we are here introducing detailed spatial simulations as a new approach in analysing such data.

Modeling 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. After a cue from fertilization, cortical actomyosin contracts toward 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.