PLoS Computational Biology: New Articles

Alternative Splicing in the Differentiation of Human Embryonic Stem Cells into Cardiac Precursors

2009-11-06T08:00:00Z

Author Summary

The reprogramming of pluripotent stem cells from adult cells is a crucial step toward producing patient-specific cells for transplant therapy. Critical to this goal is the ability to reproducibly drive the differentiation of these cells to specific fates, such as cardiac and neural cells. While gene expression is important in tissue specific differentiation, the impact of alternative splicing on the biology of differentiating cells has not been fully realized. To identify specific splicing events that may determine cell-type-specific differentiation, we compared splicing profiles of human embryonic stem cells (ESCs) and derived cardiac and neural precursors using Affymetrix exon tiling arrays. Segregation of splicing profiles into cardiac-restricted and common cardiac/neural differentiation pattern groups revealed unique groups of genes with clear implications for the biology of cardiomyocyte function and the maintenance of pluripotent ESCs. Alternative splicing of many of these genes, notably regulators of cell death and proliferation, were often predicted to impact protein domain or microRNA binding site inclusion, suggesting that the function or expression of these proteins is altered during differentiation. These results provide further evidence that alternative splicing is important in shaping the functional repertoire of ESCs and differentiated cells.

Robust Models for Optic Flow Coding in Natural Scenes Inspired by Insect Biology

2009-11-06T08:00:00Z

Author Summary

Building artificial vision systems that work robustly in a variety of environments has been difficult, with systems often only performing well under restricted conditions. In contrast, animal vision operates effectively under extremely variable situations. Many attempts to emulate biological vision have met with limited success, often because multiple seemingly appropriate approximations to neural coding resulted in a compromised system. We have constructed a full model for motion processing in the insect visual pathway incorporating known or suspected elements in as much detail as possible. We have found that it is only once all elements are present that the system performs robustly, with reduction or removal of elements dramatically limiting performance. The implementation of this new algorithm could provide a very useful and robust velocity estimator for artificial navigation systems.

A Hidden Markov Model for Single Particle Tracks Quantifies Dynamic Interactions between LFA-1 and the Actin Cytoskeleton

2009-11-06T08:00:00Z

Author Summary

Many important biological processes begin when a target molecule binds to a cell surface receptor protein. This event leads to a series of biochemical reactions involving the receptor and signalling molecules, and ultimately a cellular response. Surface receptors are mobile on the cell surface and their mobility is influenced by their interaction with intracellular proteins. We wish to understand the details of these interactions and how they are affected by cellular activation. An experimental technique called single particle tracking (SPT) uses optical microscopy to study the motion of cell-surface receptors, revealing important details about the organization of the cell membrane. In this paper, we propose a new method of analyzing SPT data to identify reduced receptor mobility as a result of transient binding to intracellular proteins. Using our analysis we are able to reliably differentiate receptor motion when a receptor is freely diffusing on the membrane versus when it is interacting with an intracellular protein. By observing the frequency of transitions between free and bound states, we are able to estimate reaction rates for the interaction. We apply our method to the receptor LFA-1 in T cells and draw conclusions about its interactions with the T cell cytoskeleton.

Evolution of Resistance to Targeted Anti-Cancer Therapies during Continuous and Pulsed Administration Strategies

2009-11-06T08:00:00Z

Author Summary

Recently, the field of anti-cancer therapy has witnessed a revolution by the discovery of targeted therapy, which refers to compounds targeting specific pathways causing abnormal growth of cancer cells. The clinical success of such drugs has been limited by the evolution of acquired resistance to these compounds, which leads to a relapse after initial response to therapy. Current dosing procedures are not designed to optimally delay the emergence of resistance; the identification of such optimal dosing schedules represents an important challenge in clinical cancer research. Here, we design a novel methodology to identify the optimum drug administration strategies that reach this clinical goal. Our model describes the evolutionary dynamics of a tumor cell population during therapy. We consider drug resistance emerging due to a single (epi)genetic alteration and calculate the probability of resistance arising during specific dosing strategies. We then optimize treatment protocols such that the risk of resistance is minimal while considering drug toxicity and side effects as constraints. Since this methodology can be extended to describe situations arising during administration of cytotoxic chemotherapy as well, it can be used to identify optimum drug administration schedules to avoid resistance for any cancer and treatment type.

Optimal Experimental Design for Parameter Estimation of a Cell Signaling Model

2009-11-06T08:00:00Z

Author Summary

Differential equation models of signaling processes are useful to gain a molecular and quantitative understanding of cellular information flow. Although these models are typically based on simple kinetic rules, they can often qualitatively describe the behavior of biological systems. However, in the quest to transform biomedical research into an engineering discipline, biologists face the challenge of estimating important parameters of such models from laboratory data. Measurement noise as well as the robust architecture of biological circuits are causes for large uncertainty of parameter estimates. This makes it difficult to plan informative experiments. Here, we used a computational method to predict and minimize the uncertainty of parameter estimates we would obtain from prospective experiments given a cancer-relevant signaling model. This was achieved by optimizing the concentrations and time points for adding drugs in a live-cell microscopy experiment. Our experimental results demonstrated that the advice given by this algorithm resulted in many-fold more informative data than we would obtain by repeating an intuitively planned experiment. Our study shows that significant experimental effort and time can be saved by adopting an optimal experimental design strategy for inferring relevant parameters from biomedical experiments.

Looking at Cerebellar Malformations through Text-Mined Interactomes of Mice and Humans

2009-11-06T08:00:00Z

Author Summary

We described and made publicly available the largest existing set of text-mined statements; we also presented its application to an important biological problem. We have extracted and purified two large molecular networks, one for humans and one for mouse. We characterized the data sets, described the methods we used to generate them, and presented a novel biological application of the networks to study the etiology of five cerebellum phenotypes. We demonstrated quantitatively that the development-related malformations differ in their system-level properties from degeneration-related genes. We showed that there is a high degree of overlap among the genes implicated in the developmental malformations, that these genes have a strong tendency to be highly connected within the molecular network, and that they also tend to be clustered together, forming a compact molecular network neighborhood. In contrast, the genes involved in malformations due to degeneration do not have a high degree of connectivity, are not strongly clustered in the network, and do not overlap significantly with the development related genes. In addition, taking into account the above-mentioned system-level properties and the gene-specific network interactions, we made highly confident predictions about novel genes that are likely also involved in the etiology of the analyzed phenotypes.

PLoS Computational Biology Issue Image | Vol. 5(10) October 2009

2009-10-30T07:00:00Z

Cloud topology in the yeast protein interaction network.

This figure represents the protein-protein interaction network topology of budding yeast. The outermost circle shows proteins with smaller numbers of interactions (gray dots) and proteins that interact with these are located on inner circles. Most large hubs are placed here (blue dots). Inner circles are dominated by medium size hubs (red dots) that are extensively connected to each other. This topology is mathematically similar to the router-level topology of the internet and represents networks according to highly optimized tolerance (see Hase et al., doi:10.1371/journal.pcbi.1000550).

Image Credit: Hiroaki Kitano and Takeshi Hase

Computational Biology in Colombia

2009-10-30T07:00:00Z

Getting Started in Gene Expression Microarray Analysis

2009-10-30T07:00:00Z

Investigation of the Interaction between the Large and Small Subunits of Potato ADP-Glucose Pyrophosphorylase

2009-10-30T07:00:00Z

Author Summary

ADP-glucose pyrophosphorylase (AGPase) is a key heterotetrameric allosteric enzyme involved in plant starch biosynthesis. In this study, we have applied computational and experimental methods to identify critical amino acids of the AGPase large and small subunits that interact with each other during the heterotetrameric structure formation. During the comparison of the computational with the experimental results we also noted that the backbone energy contribution of the interface residues is more important in identifying critical residues. This study will enable us to use a rational approach to obtain better assembled mutant AGPase variants and use them for the improvement of the plant yield.

Mechanical Strength of 17 134 Model Proteins and Cysteine Slipknots

2009-10-30T07:00:00Z

Author Summary

The advances in nanotechnology have allowed for manipulation of single biomolecules and determination of their elastic properties. Titin was among the first proteins studied in this way. Its unravelling by stretching requires a 204 pN force. The resistance to stretching comes mostly from a localized region known as a force clamp. In titin, the force clamp is simple as it is formed by two parallel β-strands that are sheared on pulling. Studies of a set of under a hundred proteins accomplished in the last decade have revealed a variety of the force clamps that lead to forces ranging from under 20 pN to about 500 pN. This set comprises only a tiny fraction of proteins known. Thus one needs guidance as to what proteins should be considered for specific mechanical properties. Such a guidance is provided here through simulations within simplified coarse-grained models on 17 134 proteins that are stretched at constant speed. We correlate their unravelling forces with two structure classification schemes. We identify proteins with large resistance to unravelling and characterize their force clamps. Quite a few top strength proteins owe their sturdiness to a new type of the force clamp: the cystein slipknot in which the force peak is due to dragging of a piece of the backbone through a closed ring formed by two other pieces of the backbone and two connecting disulphide bonds.

Gene Circuit Analysis of the Terminal Gap Gene huckebein

2009-10-30T07:00:00Z

Author Summary

Currently, there are two very different approaches to the study of pattern formation: Traditional developmental genetics investigates the role of particular factors in great mechanistic detail, while newly developed systems-biology methods study many factors in parallel but usually remain rather general in their conclusions. Here, we attempt to bridge the gap between the two by studying the expression pattern and function of a particular developmental gene—the terminal gap gene huckebein (hkb) in the fruit fly Drosophila melanogaster—in great quantitative detail using a systems-level approach called the gene circuit method. Gene circuits are mathematical models which allow us to reconstitute a developmental process in the computer. This allows us to study the function of the hkb gene in its wild-type regulatory context with unprecedented accuracy and resolution. Our results confirm earlier, qualitative evidence, and show that hkb plays a small, but crucial role in gap gene regulation. Understanding hkb's regulatory contributions is essential for our wider understanding of dynamic shifts in the position of gap gene expression domains which play important roles during both development and evolution.

Tipping the Balance: Robustness of Tip Cell Selection, Migration and Fusion in Angiogenesis

2009-10-30T07:00:00Z

Author Summary

Abnormal vasculature exacerbates many diseases such as cancer and diabetic retinopathy. In angiogenesis new blood vessels, headed by a migrating tip cell, sprout from pre-existing vessels in response to chemical signals. The signals are released from newly oxygen deficient tissue. The signals are known to be different in disease and are thought to cause the process of angiogenesis to progress abnormally, though the reasons for this remain unclear. Normalisation of angiogenesis has great potential as a therapeutic strategy; it has been shown to reduce metastasis and improve drug delivery in tumours. Here we focus on the behaviours of three inter-related initial angiogenic pathways associated with changes in tissue signal conditions, utilising both in silico and in vivo approaches. By the construction and implementation of a novel computational model for cell motility and signal processing we present a new theory on why angiogenesis exhibits such sensitivity to signal changes and show that the behaviour in disease is surprisingly more robust than normal functioning. This we attribute to the positive feedback of cell migration reinforcing abnormal oscillations in cell fate selection. We make the unique prediction that normalisation could be achieved by reducing cell migration alone.

Structure of Protein Interaction Networks and Their Implications on Drug Design

2009-10-30T07:00:00Z

Author Summary

Genome-wide data on interactions between proteins are now available, and networks of protein interactions are the keys to understanding diseases and finding accurate drug targets. This study revealed that the architectural properties of the backbones of protein interaction networks (PINs) were similar to those of the Internet router-level topology by using statistical analyses of genome-wide budding yeast and human PINs. This type of network is known as a highly optimized tolerance (HOT) network that is robust against failures in its components and that ensures high levels of communication. Moreover, we also found that a large number of the most successful drug-target proteins are on the backbone of the human PIN. We made a list of proteins on the backbone of the human PIN, which may help drug companies to search more efficiently for new drug targets.

Specific Entrainment of Mitral Cells during Gamma Oscillation in the Rat Olfactory Bulb

2009-10-30T07:00:00Z

Author Summary

Olfactory function relies on a chain of neural relays that extends from the periphery to the central nervous system and implies neural activity with various timescales. A central question in neuroscience is how information is encoded by the neural activity. In the mammalian olfactory bulb, local neural activity oscillations in the 40–80 Hz range (gamma) may influence the timing of individual neuron activities such that olfactory information may be encoded in this way. In this study, we first characterize in vivo the detailed activity of individual neurons relative to the oscillation and find that, depending on their state, neurons can exhibit periodic activity patterns. We also find, at least qualitatively, a relation between this activity and a particular odor. This is reminiscent of general physical phenomena—the entrainment by an oscillation—and to verify this hypothesis, in a second phase, we build a biologically realistic model mimicking these in vivo conditions. Our model confirms quantitatively this hypothesis and reveals that entrainment is maximal in the gamma range. Taken together, our results suggest that the neuronal activity may be specifically formatted in time during the gamma oscillation in such a way that it could, at this stage, encode the odor.

Intrinsic Structural Disorder Confers Cellular Viability on Oncogenic Fusion Proteins

2009-10-30T07:00:00Z

Author Summary

Chromosomal translocations generate chimeric proteins by fusing segments of two distinct genes and are frequently associated with cancer. The proteins involved are large and fairly heterogeneous in sequence and typically have only a few dispersed structural domains connected by long uncharacterized regions. It has never been studied from a structural perspective how these chimeras survive losing significant portions of the original proteins and acquire new oncogenic functions. By analyzing a collection of 406 human translocation proteins we show here that the answer to both questions lies to a large extent in the high level of structural disorder in the fusion partner proteins (on average, they are twice as disordered as all human proteins). The translocation breakpoints usually avoid globular domains. In rare cases when a globular domain is truncated by the fusion, it happens at a location in the domain where the hydrophobicity exposed by the split is favorable (i.e., not too high). Disorder on average is significantly higher in the vicinity of the breakpoint than in the rest of the fusion proteins. Disorder also plays a pivotal role in the acquired oncogenic function by bringing distant/disparate fusion segments together that enables novel intra- and/or intermolecular interactions.

Computational Resources in Infectious Disease: Limitations and Challenges

2009-10-26T07:00:00Z

Discovering the Phylodynamics of RNA Viruses

2009-10-26T07:00:00Z

The Role of Medical Structural Genomics in Discovering New Drugs for Infectious Diseases

2009-10-26T07:00:00Z

A Model of Cardiovascular Disease Giving a Plausible Mechanism for the Effect of Fractionated Low-Dose Ionizing Radiation Exposure

2009-10-23T07:00:00Z

Author Summary

Atherosclerosis is the main cause of coronary heart disease and stroke, the two major causes of death in developed society. There is emerging evidence of excess risk of cardiovascular disease in various occupationally exposed groups, exposed to fractionated radiation doses with small doses/fraction. The mechanisms for such effects of fractionated low-dose radiation exposures on cardiovascular disease are unclear. We outline a spatial reaction-diffusion model for early stage atherosclerotic lesion formation and perform a stability analysis, based on experimentally derived parameters. We show that following multiple small radiation doses the chemo-attractant (MCP-1) concentration increases proportionally to cumulative dose; this is driven by radiation-induced monocyte death. This will result in risk of atherosclerosis increasing approximately linearly with cumulative dose. This proposed mechanism would be testable. If true, it also has substantive implications for radiological protection, which at present does not take cardiovascular disease into account. The major uncertainty in assessing low-dose risk of cardiovascular disease is the shape of the dose response relationship, which is unclear in high dose data. Our analysis suggests that linear extrapolation would be appropriate.

Steps in the Bacterial Flagellar Motor

2009-10-23T07:00:00Z

Author Summary

Many species of bacteria swim to find food or to avoid toxins. Swimming motility depends on helical flagella that act as propellers. Each flagellum is driven by a rotary molecular engine–the bacterial flagellar motor–which draws its energy from an ion flux entering the cell. Despite much progress, the detailed mechanisms underlying the motor's extraordinary power output, as well as its near 100% efficiency, have yet to be understood. Surprisingly, recent experiments have shown that, at low speeds, the motor proceeds by small steps (~26 per rotation), providing new insight into motor operation. Here we show that a simple physical model can quantitatively account for this stepping behavior as well as the motor's near-perfect efficiency and many other known properties of the motor. In our model, torque is generated via protein-springs that pull on the rotor; the steps arise from contact forces between static components of the motor and a 26-fold periodic ring that forms part of the rotor. Our model allows us to explain some curious properties of the motor, including the observation that backward steps are shorter on average than forward steps, and to make novel, experimentally testable predictions on the motor's speed and diffusion properties.

An Atlas of the Thioredoxin Fold Class Reveals the Complexity of Function-Enabling Adaptations

2009-10-23T07:00:00Z

Author Summary

For any large class of proteins, far more protein sequences are known than can be examined experimentally. This is the case with the thioredoxin fold class, a large and diverse collection of proteins, some of which are known to catalyze important steps in metabolism. Some others participate in key processes like protein folding and detoxification of foreign compounds. Many of the unstudied proteins likely participate in other important biological processes and have useful applications in medicine and industry. We used a new network-based computational approach to create similarity-based maps of the thioredoxin fold class. These maps juxtapose unstudied proteins with similar well-characterized proteins, helping to show where existing knowledge can help predict properties of uncharacterized sequences. This information can be used to identify which of these sequences are interesting and deserve experimental characterization. We also used the maps to gain insight about how shared structural features are used and modified to affect catalysis in the different subclasses, leading to a better understanding of the interplay between structure and function in the thioredoxin fold class.

Perturbation-Response Scanning Reveals Ligand Entry-Exit Mechanisms of Ferric Binding Protein

2009-10-23T07:00:00Z

Author Summary

Upon binding ligands, many proteins undergo structural changes compared to the unbound form. We introduce a methodology to monitor these changes and to study which mechanisms arrange conformational shifts between the liganded and free forms. Our method is simple, yet it efficiently characterizes the response of proteins to a given perturbation on systematically selected residues. The coherent responses predicted are validated by molecular dynamics simulations. The results indicate that the iron uptake by the ferric binding protein is favorable in a thermally fluctuating environment, while release of iron is allosterically moderated. Since ferric binding protein exhibits a high sequence identity with human transferrin whose allosteric anion binding sites generate large conformational changes around the binding region, we suggest mutational studies on remotely controlling sites identified in this work.

Invariant Distribution of Promoter Activities in Escherichia coli

2009-10-23T07:00:00Z

Author Summary

Cells respond to a changing environment by regulating the activity of genes. Here, we sought to understand how E. coli cells distribute their limited transcriptional resources among their target genes, and how this allocation varies with growth rate and growth conditions. To achieve this, we assayed the expression of a comprehensive library of transcriptional reporter strains under different conditions. High-temporal resolution measurements of promoter activities were obtained for different growth rates spanning recovery from stationary phase into exponential phase and eventually deep stationary phase again. We find that the genome-wide promoter activity follows a power-law distribution, which depends solely on growth rate and is independent of the specific growth conditions. Moreover, we find that the power-law distribution can be decomposed into two log-normal distributions: metabolic promoters that make up the low end of the distribution, and ribosomal promoters that make up the high end of the distribution. While distributions remained constant for a given growth rate, the ranked expression of metabolic promoters differed according to the specific condition. Thus, the invariant distribution may suggest optimal resource allocation under constrained resources. A mathematical theory is presented to explain these results.

Modeling Latently Infected Cell Activation: Viral and Latent Reservoir Persistence, and Viral Blips in HIV-infected Patients on Potent Therapy

2009-10-16T07:00:00Z

Author Summary

Current combination therapy can suppress viral loads in HIV-1-infected individuals to below the detection limit of standard commercial assays. However, it cannot eradicate the virus from patients. HIV-1 can generally be identified in resting memory CD4⁺ T cells and persists in patients on potent treatment for a long time. These latently infected cells decay slowly, but can produce new virions when activated by relevant antigens. Many patients experience transient episodes of viremia, or blips, even though they have “undetectable” plasma viral loads for many years. Here, we develop a new mathematical model describing latently infected cell activation upon random antigenic stimulation. Using the model, we show that programmed expansion and contraction of latently infected cells upon activation can generate both low viral load persistence and viral blips. Occasional replenishment of the latent reservoir may explain the different decay kinetics of the reservoir observed in clinical practice. We also show that a model with homeostatic proliferation of latently infected cells can explain persistence of low-level virus, stability of the latent reservoir, and emergence of viral blips. These results provide novel insights into the long-term virus dynamics and could have implications for the treatment of HIV-1 infection.

‘Glocal’ Robustness Analysis and Model Discrimination for Circadian Oscillators

2009-10-16T07:00:00Z

Author Summary

Robustness is an intrinsic property of many biological systems. To quantify the robustness of a model that represents such a system, two approaches exist: global methods assess the volume in parameter space that is compliant with the proper functioning of the system; and local methods, in contrast, study the model for a given parameter set and determine its robustness. Local methods are fundamentally biased due to the a priori choice of a particular parameter set. Our ‘glocal’ analysis combines the two complementary approaches and provides an objective measure of robustness. We apply this method to two prominent, recent models of the cyanobacterial circadian oscillator. Our results allow discriminating the two models based on this analysis: both global and local measures of robustness favor one of the two models. The ‘glocal’ method also identifies key factors that influence robustness. For instance, we find that in both models the most fragile reactions are the ones that affect the concentration of the feedback component.

Antigenic Diversity, Transmission Mechanisms, and the Evolution of Pathogens

2009-10-16T07:00:00Z

Author Summary

Infectious diseases vary widely in how they affect those who get infected and how they are transmitted. As an example, the duration of a single infection can range from days to years, while transmission can occur via the respiratory route, water or sexual contact. Measles and HIV are contrasting examples—both are caused by RNA viruses, but one is a genetically diverse, lethal sexually transmitted infection (STI) while the other is a relatively mild respiratory childhood disease with low antigenic diversity. We investigate why the most transmissible respiratory diseases such as measles and rubella are antigenically static, meaning immunity is lifelong, while other diseases—such as influenza, or the sexually transmitted diseases—seem to trade transmissibility for the ability to generate multiple diverse strains so as to evade host immunity. We use mathematical models of disease progression and evolution within the infected host coupled with models of transmission between hosts to explore how transmission modes, host contact rates and network structure determine antigenic diversity, infectiousness and duration of infection. In doing so, we classify infections into three types—measles-like (high transmissibility, but antigenically static), flu-like (lower transmissibility, but more antigenically diverse), and STI-like (very antigenically diverse, long lived infection, but low overall transmissibility).

Subbarrel Patterns in Somatosensory Cortical Barrels Can Emerge from Local Dynamic Instabilities

2009-10-16T07:00:00Z

Author Summary

Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In rodent somatosensory cortex, groups of neurons called “barrels” correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. We suggest that these so-called subbarrel patterns arise spontaneously during development through a pattern-forming instability. We use a simple chemotaxis and branching model to explain the patterns and their dependence on the size of the barrel.

Grasping Objects with Environmentally Induced Position Uncertainty

2009-10-16T07:00:00Z

Author Summary

Optimal sensorimotor control models actions as decisions that maximize the desirableness of outcomes, where the desirableness is captured by an expected cost or utility to each action sequence. These models provide explanations for many aspects of our ability to compensate for uncertainty, but they have not been applied to understanding purposive movements—movements involving the application of forces to change the relative position of objects and the actor in the environment. Using time efficiency as a natural cost function, we present a statistical optimal control analysis of uncertainty compensation strategies in a purposive movement task, grasping an object with directional position uncertainty. In accord with the predictions of the analysis, the experimental results showed that people compensate for uncertainty by adopting grasp strategies that increase the chance to produce a stable grasp at first contact. Our findings suggest that visuomotor system plans for uncertainty even in complex purposive movements.

Multilevel Selection in Models of Prebiotic Evolution II: A Direct Comparison of Compartmentalization and Spatial Self-Organization

2009-10-16T07:00:00Z

Author Summary

The origin of life has ever been attracting scientific inquiries. The RNA world hypothesis suggests that, before the evolution of DNA and protein, primordial life was based on RNA-like molecules both for information storage and chemical catalysis. In the simplest form, an RNA world consists of RNA molecules that can catalyze the replication of their own copies. Thus, an interesting question is whether a system of RNA-like replicators can increase its complexity through Darwinian evolution and approach the modern form of life. It is, however, known that simple natural selection acting on individual replicators is insufficient to account for the evolution of complexity due to the evolution of parasite-like templates. Two solutions have been suggested: compartmentalization of replicators by membranes (i.e., protocells) and spatial self-organization of a replicator population. Here, we make a direct comparison of the two suggestions by computer simulations. Our results show that the two suggestions can lead to unanticipated and contrasting consequences in the long-term evolution of replicating molecules. The results also imply a novel advantage in the spatial self-organization for the evolution of complexity in RNA-like replicator systems.