Best practices for the execution, analysis, and data storage of plant single-cell/nucleus transcriptomics

Single-cell and single-nucleus RNA-sequencing technologies capture the expression of plant genes at an unprecedented resolution. Therefore, these technologies are gaining traction in plant molecular and developmental biology for elucidating the transcriptional changes across cell types in a specific tissue or organ, upon treatments, in response to biotic and abiotic stresses, or between genotypes. Despite the rapidly accelerating use of these technologies, collective and standardized experimental and analytical procedures to support the acquisition of high-quality data sets are still missing. In this commentary, we discuss common challenges associated with the use of single-cell transcriptomics in plants and propose general guidelines to improve reproducibility, quality, comparability, and interpretation and to make the data readily available to the community in this fast-developing field of research.

Biologically meaningful regulatory logic enhances the convergence rate in Boolean networks and bushiness of their state transition graph

Boolean models of gene regulatory networks (GRNs) have gained widespread traction as they can easily recapitulate cellular phenotypes via their attractor states. Their overall dynamics are embodied in a state transition graph (STG). Indeed, two Boolean networks (BNs) with the same network structure and attractors can have drastically different STGs depending on the type of Boolean functions (BFs) employed. Our objective here is to systematically delineate the effects of different classes of BFs on the structural features of the STG of reconstructed Boolean GRNs while keeping network structure and biological attractors fixed, and explore the characteristics of BFs that drive those features. Using $10$ reconstructed Boolean GRNs, we generate ensembles that differ in BFs and compute from their STGs the dynamics' rate of contraction or 'bushiness' and rate of 'convergence', quantified with measures inspired from cellular automata (CA) that are based on the garden-of-Eden (GoE) states. We find that biologically meaningful BFs lead to higher STG 'bushiness' and 'convergence' than random ones. Obtaining such 'global' measures gets computationally expensive with larger network sizes, stressing the need for feasible proxies. So we adapt Wuensche's $Z$-parameter in CA to BFs in BNs and provide four natural variants, which, along with the average sensitivity of BFs computed at the network level, comprise our descriptors of local dynamics and we find some of them to be good proxies for bushiness. Finally, we provide an excellent proxy for the 'convergence' based on computing transient lengths originating at random states rather than GoE states.

Preponderance of generalized chain functions in reconstructed Boolean models of biological networks

Boolean networks (BNs) have been extensively used to model gene regulatory networks (GRNs). The dynamics of BNs depend on the network architecture and regulatory logic rules (Boolean functions (BFs)) associated with nodes. Nested canalyzing functions (NCFs) have been shown to be enriched among the BFs in the large-scale studies of reconstructed Boolean models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. We build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose two other sub-types of NCFs, namely, the class of chain-1 functions and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 (also holds for chain-1) functions decreases exponentially with the number of inputs. We provide analytical treatment for this and other observations on BFs. Then, by analyzing three different datasets of reconstructed Boolean models we find that generalized chain functions are significantly enriched within the NCFs. Lastly we illustrate that upon imposing the constraints of generalized chain functions on three different GRNs we are able to obtain biologically viable Boolean models.

Leveraging developmental landscapes for model selection in Boolean gene regulatory networks

Boolean models are a well-established framework to model developmental gene regulatory networks (DGRNs) for acquisition of cellular identities. During the reconstruction of Boolean DGRNs, even if the network structure is given, there is generally a large number of combinations of Boolean functions that will reproduce the different cell fates (biological attractors). Here we leverage the developmental landscape to enable model selection on such ensembles using the relative stability of the attractors. First we show that previously proposed measures of relative stability are strongly correlated and we stress the usefulness of the one that captures best the cell state transitions via the mean first passage time (MFPT) as it also allows the construction of a cellular lineage tree. A property of great computational importance is the insensitivity of the different stability measures to changes in noise intensities. That allows us to use stochastic approaches to estimate the MFPT and thereby scale up the computations to large networks. Given this methodology, we revisit different Boolean models of Arabidopsis thaliana root development, showing that a most recent one does not respect the biologically expected hierarchy of cell states based on relative stabilities. We therefore developed an iterative greedy algorithm that searches for models which satisfy the expected hierarchy of cell states and found that its application to the root development model yields many models that meet this expectation. Our methodology thus provides new tools that can enable reconstruction of more realistic and accurate Boolean models of DGRNs.

SeSAM: software for automatic construction of order-robust linkage maps

Background

Genotyping and sequencing technologies produce increasingly large numbers of genetic markers with potentially high rates of missing or erroneous data. Therefore, the construction of linkage maps is more and more complex. Moreover, the size of segregating populations remains constrained by cost issues and is less and less commensurate with the numbers of SNPs available. Thus, guaranteeing a statistically robust marker order requires that maps include only a carefully selected subset of SNPs.

Results

In this context, the SeSAM software allows automatic genetic map construction using seriation and placement approaches, to produce (1) a high-robustness framework map which includes as many markers as possible while keeping the order robustness beyond a given statistical threshold, and (2) a high-density total map including the framework plus almost all polymorphic markers. During this process, care is taken to limit the impact of genotyping errors and of missing data on mapping quality. SeSAM can be used with a wide range of biparental populations including from outcrossing species for which phases are inferred on-the-fly by maximum-likelihood during map elongation. The package also includes functions to simulate data sets, convert data formats, detect putative genotyping errors, visualize data and map quality (including graphical genotypes), and merge several maps into a consensus. SeSAM is also suitable for interactive map construction, by providing lower-level functions for 2-point and multipoint EM analyses. The software is implemented in a R package including functions in C++.

Conclusions

SeSAM is a fully automatic linkage mapping software designed to (1) produce a framework map as robust as desired by optimizing the selection of a subset of markers, and (2) produce a high-density map including almost all polymorphic markers. The software can be used with a wide range of biparental mapping populations including cases from outcrossing. SeSAM is freely available under a GNU GPL v3 license and works on Linux, Windows, and macOS platforms. It can be downloaded together with its user-manual and quick-start tutorial from ForgeMIA (SeSAM project) at https://forgemia.inra.fr/gqe-acep/sesam/-/releases

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-022-05045-7.

Relative importance of composition structures and biologically meaningful logics in bipartite Boolean models of gene regulation

Boolean networks have been widely used to model gene networks. However, such models are coarse-grained to an extent that they abstract away molecular specificities of gene regulation. Alternatively, bipartite Boolean network models of gene regulation explicitly distinguish genes from transcription factors (TFs). In such bipartite models, multiple TFs may simultaneously contribute to gene regulation by forming heteromeric complexes, thus giving rise to composition structures. Since bipartite Boolean models are relatively recent, an empirical investigation of their biological plausibility is lacking. Here, we estimate the prevalence of composition structures arising through heteromeric complexes. Moreover, we present an additional mechanism where composition structures may arise as a result of multiple TFs binding to cis-regulatory regions and provide empirical support for this mechanism. Next, we compare the restriction in BFs imposed by composition structures and by biologically meaningful properties. We find that though composition structures can severely restrict the number of Boolean functions (BFs) driving a gene, the two types of minimally complex BFs, namely nested canalyzing functions (NCFs) and read-once functions (RoFs), are comparatively more restrictive. Finally, we find that composition structures are highly enriched in real networks, but this enrichment most likely comes from NCFs and RoFs.

Minimum complexity drives regulatory logic in Boolean models of living systems

The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd "bias", that is they produce "on" outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

A preference for link operator functions can drive Boolean biological networks towards critical dynamics

Boolean modelling is a powerful framework to understand the operating principles of biological networks. The regulatory logic between biological entities in these networks is expressed as Boolean functions (BFs). There exist various types of BFs (and thus regulatory logic rules) which are meaningful in the biological context. In this contribution, we explore one such type, known as link operator functions (LOFs). We theoretically enumerate these functions and show that, among all BFs and even within the biologically consistent effective and unate functions (EUFs), the LOFs form a tiny subset. We then find that the AND-NOT LOFs are particularly abundant in reconstructed biological Boolean networks. By leveraging these facts, namely, the tiny representation of LOFs in the space of EUFs and their presence in the biological dataset, we show that the space of acceptable models can be shrunk by applying steady-state constraints to BFs, followed by the choice of LOFs which satisfy those constraints. Finally, we demonstrate that among a wide range of BFs, the LOFs drive biological network dynamics towards criticality.

Quantitative modelling of fine-scale variations in the Arabidopsis thaliana crossover landscape

In, essentially, all species where meiotic crossovers (COs) have been studied, they occur preferentially in open chromatin, typically near gene promoters and to a lesser extent, at the end of genes. Here, in the case of Arabidopsis thaliana, we unveil further trends arising when one considers contextual information, namely summarised epigenetic status, gene or intergenic region size, and degree of divergence between homologs. For instance, we find that intergenic recombination rate is reduced if those regions are less than 1.5 kb in size. Furthermore, we propose that the presence of single nucleotide polymorphisms enhances the rate of CO formation compared to when homologous sequences are identical, in agreement with previous works comparing rates in adjacent homozygous and heterozygous blocks. Lastly, by integrating these different effects, we produce a quantitative and predictive model of the recombination landscape that reproduces much of the experimental variation.

Enhancing backcross programs through increased recombination

Background

Introgression of a quantitative trait locus (QTL) by successive backcrosses is used to improve elite lines (recurrent parent) by introducing alleles from exotic material (donor parent). In the absence of selection, the proportion of the donor genome decreases by half at each generation. However, since selection is for the donor allele at the QTL, elimination of the donor genome around that QTL will be much slower than in the rest of the genome (i.e. linkage drag). Using markers to monitor the genome around the QTL and in the genetic background can accelerate the return to the recurrent parent genome. Successful introgression of a locus depends partly on the occurrence of crossovers at favorable positions. However, the number of crossovers per generation is limited and their distribution along the genome is heterogeneous. Recently, techniques have been developed to modify these two recombination parameters.

Results

In this paper, we assess, by simulations in the context of Brassicaceae, the effect of increased recombination on the efficiency of introgression programs by studying the decrease in linkage drag and the recovery of the recurrent genome. The simulated selection schemes begin by two generations of foreground selection and continue with one or more generations of background selection. Our results show that, when the QTL is in a region that initially lacked crossovers, an increase in recombination rate can decrease linkage drag by nearly ten-fold after the foreground selection and improves the return to the recurrent parent. However, if the QTL is in a region that is already rich in crossovers, an increase in recombination rate is detrimental.

Conclusions

Depending on the recombination rate in the region targeted for introgression, increasing it can be beneficial or detrimental. Thus, the simulations analysed in this paper help us understand how an increase in recombination rate can be beneficial. They also highlight the best methods that can be used to increase recombination rate, depending on the situation.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12711-021-00619-0.

Plant roots sense soil compaction through restricted ethylene diffusion

Soil compaction represents a major challenge for modern agriculture. Compaction is intuitively thought to reduce root growth by limiting the ability of roots to penetrate harder soils. We report that root growth in compacted soil is instead actively suppressed by the volatile hormone ethylene. We found that mutant Arabidopsis and rice roots that were insensitive to ethylene penetrated compacted soil more effectively than did wild-type roots. Our results indicate that soil compaction lowers gas diffusion through a reduction in air-filled pores, thereby causing ethylene to accumulate in root tissues and trigger hormone responses that restrict growth. We propose that ethylene acts as an early warning signal for roots to avoid compacted soils, which would be relevant to research into the breeding of crops resilient to soil compaction.

CNVmap: A Method and Software To Detect and Map Copy Number Variants from Segregation Data

Single nucleotide polymorphisms (SNPs) are used widely for detecting quantitative trait loci, or for searching for causal variants of diseases. Nevertheless, structural variations such as copy-number variants (CNVs) represent a large part of natural genetic diversity, and contribute significantly to trait variation. Numerous methods and softwares based on different technologies (amplicons, CGH, tiling, or SNP arrays, or sequencing) have already been developed to detect CNVs, but they bypass a wealth of information such as genotyping data from segregating populations, produced, e.g., for QTL mapping. Here, we propose an original method to both detect and genetically map CNVs using mapping panels. Specifically, we exploit the apparent heterozygous state of duplicated loci: peaks in appropriately defined genome-wide allelic profiles provide highly specific signatures that identify the nature and position of the CNVs. Our original method and software can detect and map automatically up to 33 different predefined types of CNVs based on segregation data only. We validate this approach on simulated and experimental biparental mapping panels in two maize populations and one wheat population. Most of the events found correspond to having just one extra copy in one of the parental lines, but the corresponding allelic value can be that of either parent. We also find cases with two or more additional copies, especially in wheat, where these copies locate to homeologues. More generally, our computational tool can be used to give additional value, at no cost, to many datasets produced over the past decade from genetic mapping panels.

Assessing by Modeling the Consequences of Increased Recombination in Recurrent Selection of Oryza sativa and Brassica rapa

Meiotic recombination generates genetic diversity but in most species the number of crossovers per meiosis is limited. Previous modeling studies showed that increasing recombination can enhance response to selection. However, such studies did not assume a specific method of modifying recombination. Our objective was to test whether two methods used to increase recombination in plants could increase genetic gain in a population undergoing recurrent selection such as in genomic selection programs. The first method, in Oryza sativa, used a mutant of anti-crossover genes, increasing global recombination without affecting the recombination landscape shape. The second one used the ploidy level of a cross between Brassica rapa and Brassica napus, increasing recombination especially in pericentromeric regions. Our modeling framework used these recombination landscapes and sampled quantitative trait loci positions from the actual gene distributions. We simulated selection programs with initially a cross between two inbred lines, for two species. Increased recombination enhanced the response to selection. The amount of enhancement in the cumulative gain largely depended on the species and the number of quantitative trait loci (2, 10, 20, 50, 200 or 1000 per chromosome). Genetic gains were increased up to 30% after 20 generations. Furthermore, increasing recombination in cold regions was the most effective: the gain was larger by 25% with the first method and 34% with the second one in B. rapa, and 12% compared to 16% in O. sativa In summary, increased recombination enhances the genetic gain in long-term selection programs, with visible effects after four to five generations.

Recombination suppression in heterozygotes for a pericentric inversion induces the interchromosomal effect on crossovers in Arabidopsis

During meiosis, recombination ensures allelic exchanges through crossovers (COs) between the homologous chromosomes. Advances in our understanding of the rules of COs have come from studies of mutations including structural chromosomal rearrangements that, when heterozygous, are known to impair COs in various organisms. In this work, we investigate the effect of a large heterozygous pericentric inversion on male and female recombination in Arabidopsis. The inversion was discovered in the Atmcc1 mutant background and was characterized through genetic and next-generation sequencing analysis. Reciprocal backcross populations, each consisting of over 400 individuals, obtained from the mutant and the wild type, both crossed with Landsberg erecta, were analyzed genome-wide by 143 single-nucleotide polymorphisms. The negative impact of inversion became evident in terms of CO loss in the rearranged chromosome in both male and female meiosis. No single-CO event was detected within the inversion, consistent with a post-meiotic selection operating against unbalanced gametes. Cytological analysis of chiasmata in F₁ plants confirmed that COs were reduced in male meiosis in the chromosome with inversion. Crossover suppression on the rearranged chromosome is associated with a significant increase of COs in the other chromosomes, thereby maintaining unchanged the number of COs per cell. The CO pattern observed in our study is consistent with the interchromosomal (IC) effect as first described in Drosophila. In contrast to male meiosis, in female meiosis no IC effect is visible. This may be related to the greater strength of interference that constrains the CO number in excess of the minimum value imposed by CO assurance in Arabidopsis female meiosis.

Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines

Recombinant Inbred Lines (RILs) are obtained through successive generations of inbreeding. In 1931 Haldane and Waddington published a landmark paper where they provided the probabilities of achieving any combination of alleles in 2-way RILs for 2 and 3 loci. In the case of sibling RILs where sisters and brothers are crossed at each generation, there has been no progress in treating 4 or more loci, a limitation we overcome here without much increase in complexity. In the general situation of L loci, the task is to determine 2 ^L probabilities, but we find that it is necessary to first calculate the 4 ^L "identical by descent" (IBD) probabilities that a RIL inherits at each locus its DNA from one of the four originating chromosomes. We show that these 4 ^L probabilities satisfy a system of linear equations that follow from self-consistency. In the absence of genetic interference-crossovers arising independently-the associated matrix can be written explicitly in terms of the recombination rates between the different loci. We provide the matrices for L up to 4 and also include a computer program to automatically generate the matrices for higher values of L. Furthermore, our framework can be generalized to recombination rates that are different in female and male meiosis which allows us to show that the Haldane and Waddington 2-locus formula is valid in that more subtle case if the meiotic recombination rate is taken as the average rate across female and male. Once the 4 ^L IBD probabilities are determined, the 2 ^L probabilities of RIL genotypes are obtained via summations of these quantities. In fine, our computer program allows to determine the probabilities of all the multilocus genotypes produced in such sibling-based RILs for L<=10, a huge leap beyond the L = 3 restriction of Haldane and Waddington.

Feedback between environment and traits under selection in a seasonal environment: consequences for experimental evolution

Batch cultures are frequently used in experimental evolution to study the dynamics of adaptation. Although they are generally considered to simply drive a growth rate increase, other fitness components can also be selected for. Indeed, recurrent batches form a seasonal environment where different phases repeat periodically and different traits can be under selection in the different seasons. Moreover, the system being closed, organisms may have a strong impact on the environment. Thus, the study of adaptation should take into account the environment and eco-evolutionary feedbacks. Using data from an experimental evolution on yeast Saccharomyces cerevisiae, we developed a mathematical model to understand which traits are under selection, and what is the impact of the environment for selection in a batch culture. We showed that two kinds of traits are under selection in seasonal environments: life-history traits, related to growth and mortality, but also transition traits, related to the ability to react to environmental changes. The impact of environmental conditions can be summarized by the length of the different seasons which weight selection on each trait: the longer a season is, the higher the selection on associated traits. Since phenotypes drive season length, eco-evolutionary feedbacks emerge. Our results show how evolution in successive batches can affect season lengths and strength of selection on different traits.

High-throughput measurement of recombination rates and genetic interference in Saccharomyces cerevisiae

Allelic recombination owing to meiotic crossovers is a major driver of genome evolution, as well as a key player for the selection of high-performing genotypes in economically important species. Therefore, we developed a high-throughput and low-cost method to measure recombination rates and crossover patterning (including interference) in large populations of the budding yeast Saccharomyces cerevisiae. Recombination and interference were analysed by flow cytometry, which allows time-consuming steps such as tetrad microdissection or spore growth to be avoided. Moreover, our method can also be used to compare recombination in wild-type vs. mutant individuals or in different environmental conditions, even if the changes in recombination rates are small. Furthermore, meiotic mutants often present recombination and/or pairing defects affecting spore viability but our method does not involve growth steps and thus avoids filtering out non-viable spores.

Short relaxation times but long transient times in both simple and complex reaction networks

When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time τ, i.e. the inverse of the decay rate, is the longest characteristic time in the system. However, that need not be true, other times such as the lifetime of an infinitesimal perturbation can be much longer. In the present work, we demonstrate that this paradoxical property can arise even in quite simple systems such as a linear chain of reactions obeying mass action (MA) kinetics. By mathematical analysis of simple reaction networks, we pin-point the reason why the standard relaxation time does not provide relevant information on the potentially long transient times of typical infinitesimal perturbations. Overall, we consider four characteristic times and study their behaviour in both simple linear chains and in more complex reaction networks taken from the publicly available database ‘Biomodels’. In all these systems, whether involving MA rates, Michaelis–Menten reversible kinetics, or phenomenological laws for reaction rates, we find that the characteristic times corresponding to lifetimes of tracers and of concentration perturbations can be significantly longer than τ.

Amplifying recombination genome-wide and reshaping crossover landscapes in Brassicas

Meiotic recombination by crossovers (COs) is tightly regulated, limiting its key role in producing genetic diversity. However, while COs are usually restricted in number and not homogenously distributed along chromosomes, we show here how to disrupt these rules in Brassica species by using allotriploid hybrids (AAC, 2n = 3x = 29), resulting from the cross between the allotetraploid rapeseed (B. napus, AACC, 2n = 4x = 38) and one of its diploid progenitors (B. rapa, AA, 2n = 2x = 20). We produced mapping populations from different genotypes of both diploid AA and triploid AAC hybrids, used as female and/or as male. Each population revealed nearly 3,000 COs that we studied with SNP markers well distributed along the A genome (on average 1 SNP per 1.25 Mbp). Compared to the case of diploids, allotriploid hybrids showed 1.7 to 3.4 times more overall COs depending on the sex of meiosis and the genetic background. Most surprisingly, we found that such a rise was always associated with (i) dramatic changes in the shape of recombination landscapes and (ii) a strong decrease of CO interference. Hybrids carrying an additional C genome exhibited COs all along the A chromosomes, even in the vicinity of centromeres that are deprived of COs in diploids as well as in most studied species. Moreover, in male allotriploid hybrids we found that Class I COs are mostly responsible for the changes of CO rates, landscapes and interference. These results offer the opportunity for geneticists and plant breeders to dramatically enhance the generation of diversity in Brassica species by disrupting the linkage drag coming from limits on number and distribution of COs.

Phenotypic innovation through recombination in genome-scale metabolic networks

Recombination is an important source of metabolic innovation, especially in prokaryotes, which have evolved the ability to survive on many different sources of chemical elements and energy. Metabolic systems have a well-understood genotype-phenotype relationship, which permits a quantitative and biochemically principled understanding of how recombination creates novel phenotypes. Here, we investigate the power of recombination to create genome-scale metabolic reaction networks that enable an organism to survive in new chemical environments. To this end, we use flux balance analysis, an experimentally validated computational method that can predict metabolic phenotypes from metabolic genotypes. We show that recombination is much more likely to create novel metabolic abilities than random changes in chemical reactions of a metabolic network. We also find that phenotypic innovation is more likely when recombination occurs between parents that are genetically closely related, phenotypically highly diverse, and viable on few rather than many carbon sources. Survival on a new carbon source preferentially involves reactions that are superessential, that is, essential in many metabolic networks. We validate our observations with data from 61 reconstructed prokaryotic metabolic networks. Our systematic and quantitative analysis of metabolic systems helps understand how recombination creates innovation.

Constrained Allocation Flux Balance Analysis

New experimental results on bacterial growth inspire a novel top-down approach to study cell metabolism, combining mass balance and proteomic constraints to extend and complement Flux Balance Analysis. We introduce here Constrained Allocation Flux Balance Analysis, CAFBA, in which the biosynthetic costs associated to growth are accounted for in an effective way through a single additional genome-wide constraint. Its roots lie in the experimentally observed pattern of proteome allocation for metabolic functions, allowing to bridge regulation and metabolism in a transparent way under the principle of growth-rate maximization. We provide a simple method to solve CAFBA efficiently and propose an “ensemble averaging” procedure to account for unknown protein costs. Applying this approach to modeling E. coli metabolism, we find that, as the growth rate increases, CAFBA solutions cross over from respiratory, growth-yield maximizing states (preferred at slow growth) to fermentative states with carbon overflow (preferred at fast growth). In addition, CAFBA allows for quantitatively accurate predictions on the rate of acetate excretion and growth yield based on only 3 parameters determined by empirical growth laws.

The intracellular protein levels of exponentially growing bacteria are known to vary strongly with growth conditions, as described by quantitative “growth laws”. This work introduces a computational genome-scale framework (Constrained Allocation Flux Balance Analysis, CAFBA) which incorporates growth laws into canonical Flux Balance Analysis. Upon introducing 3 parameters based on established growth laws for E. coli, CAFBA accurately reproduces empirical results on the growth-rate dependent rate of carbon overflow and growth yield, and generates testable predictions about cellular energetic strategies and protein expression levels. CAFBA therefore provides a simple, quantitative approach to balancing the trade-off between growth and its associated biosynthetic costs at genome-scale, without the burden of tuning many inaccessible parameters.

Modelling the emergence of polarity patterns for the intercellular transport of auxin in plants

The hormone auxin is actively transported throughout plants via protein machineries including the dedicated transporter known as PIN. The associated transport is ordered with nearby cells driving auxin flux in similar directions. Here, we provide a model of both the auxin transport and of the dynamics of cellular polarization based on flux sensing. Our main findings are: (i) spontaneous intracellular PIN polarization arises if PIN recycling dynamics are sufficiently nonlinear, (ii) there is no need for an auxin concentration gradient and (iii) ordered multi-cellular patterns of PIN polarization are favoured by molecular noise.

Combined fluorescent and electron microscopic imaging unveils the specific properties of two classes of meiotic crossovers

Crossovers (COs) shuffle genetic information and allow balanced segregation of homologous chromosomes during the first division of meiosis. In several organisms, mutants demonstrate that two molecularly distinct pathways produce COs. One pathway produces class I COs that exhibit interference (lowered probability of nearby COs), and the other pathway produces class II COs with little or no interference. However, the relative contributions, genomic distributions, and interactions of these two pathways are essentially unknown in nonmutant organisms because marker segregation only indicates that a CO has occurred, not its class type. Here, we combine the efficiency of light microscopy for revealing cellular functions using fluorescent probes with the high resolution of electron microscopy to localize and characterize COs in the same sample of meiotic pachytene chromosomes from wild-type tomato. To our knowledge, for the first time, every CO along each chromosome can be identified by class to unveil specific characteristics of each pathway. We find that class I and II COs have different recombination profiles along chromosomes. In particular, class II COs, which represent about 18% of all COs, exhibit no interference and are disproportionately represented in pericentric heterochromatin, a feature potentially exploitable in plant breeding. Finally, our results demonstrate that the two pathways are not independent because there is interference between class I and II COs.

Crossover localisation is regulated by the neddylation posttranslational regulatory pathway

A genetic study finds the neddylation pathway (known to-date for post-translational protein modification) is involved in regulating crossover localization but not crossover number during meiosis in Arabidopsis.

Crossovers (COs) are at the origin of genetic variability, occurring across successive generations, and they are also essential for the correct segregation of chromosomes during meiosis. Their number and position are precisely controlled, however the mechanisms underlying these controls are poorly understood. Neddylation/rubylation is a regulatory pathway of posttranslational protein modification that is required for numerous cellular processes in eukaryotes, but has not yet been linked to homologous recombination. In a screen for meiotic recombination-defective mutants, we identified several axr1 alleles, disrupting the gene encoding the E1 enzyme of the neddylation complex in Arabidopsis. Using genetic and cytological approaches we found that axr1 mutants are characterised by a shortage in bivalent formation correlated with strong synapsis defects. We determined that the bivalent shortage in axr1 is not due to a general decrease in CO formation but rather due to a mislocalisation of class I COs. In axr1, as in wild type, COs are still under the control of the ZMM group of proteins. However, in contrast to wild type, they tend to cluster together and no longer follow the obligatory CO rule. Lastly, we showed that this deregulation of CO localisation is likely to be mediated by the activity of a cullin 4 RING ligase, known to be involved in DNA damage sensing during somatic DNA repair and mouse spermatogenesis. In conclusion, we provide evidence that the neddylation/rubylation pathway of protein modification is a key regulator of meiotic recombination. We propose that rather than regulating the number of recombination events, this pathway regulates their localisation, through the activation of cullin 4 RING ligase complexes. Possible targets for these ligases are discussed.

During meiosis, two successive chromosomal divisions follow a single S phase, resulting in the formation of four haploid cells, each with half of the parental genetic material. This reduction in chromosome number occurs during the first meiotic division, when homologous chromosomes (paternal and maternal) are separated from each other. For this to happen, homologous chromosomes associate in structures called bivalents, where each chromosome is linked to its homologue by a point of contact known as chiasmata. These chiasmata reflect the formation of crossovers (COs), one of the manifestations of the exchange of genetic material occurring during homologous recombination. CO number varies little at around two per chromosome pair, and they tend to be evenly spaced on chromosomes. Thus, CO number and distribution are very tightly controlled. However, the mechanisms underlying these controls are very poorly understood. In this study, we identified a regulatory pathway of meiotic recombination. We show that this pathway does not regulate the amount of recombination events per se, but instead controls their localisation, as when it is defective, CO events cluster together in a few regions of the genome, leading to bivalent shortage and progeny aneuploidy with incorrect numbers of chromosomes. This regulatory pathway is a posttranslational protein modification system called neddylation (or rubylation in plants), known to be required for numerous cellular processes in eukaryotes. We identify an enzyme of the neddylation complex as a major regulator of meiotic recombination in Arabidopsis and show that this process may be also conserved in mammals.

Historical contingency and the gradual evolution of metabolic properties in central carbon and genome-scale metabolisms

BACKGROUND

A metabolism can evolve through changes in its biochemical reactions that are caused by processes such as horizontal gene transfer and gene deletion. While such changes need to preserve an organism's viability in its environment, they can modify other important properties, such as a metabolism's maximal biomass synthesis rate and its robustness to genetic and environmental change. Whether such properties can be modulated in evolution depends on whether all or most viable metabolisms - those that can synthesize all essential biomass precursors - are connected in a space of all possible metabolisms. Connectedness means that any two viable metabolisms can be converted into one another through a sequence of single reaction changes that leave viability intact. If the set of viable metabolisms is disconnected and highly fragmented, then historical contingency becomes important and restricts the alteration of metabolic properties, as well as the number of novel metabolic phenotypes accessible in evolution.

RESULTS

We here computationally explore two vast spaces of possible metabolisms to ask whether viable metabolisms are connected. We find that for all but the simplest metabolisms, most viable metabolisms can be transformed into one another by single viability-preserving reaction changes. Where this is not the case, alternative essential metabolic pathways consisting of multiple reactions are responsible, but such pathways are not common.

CONCLUSIONS

Metabolism is thus highly evolvable, in the sense that its properties could be fine-tuned by successively altering individual reactions. Historical contingency does not strongly restrict the origin of novel metabolic phenotypes.

Crossover rate between homologous chromosomes and interference are regulated by the addition of specific unpaired chromosomes in Brassica

Recombination is a major mechanism generating genetic diversity, but the control of the crossover rate remains a key question. In Brassica napus (AACC, 2n = 38), we can increase the homologous recombination between A genomes in AAC hybrids. Hypotheses for this effect include the number of C univalent chromosomes, the ratio between univalents and bivalents and, finally, which of the chromosomes are univalents. To test these hypotheses, we produced AA hybrids with zero, one, three, six or nine additional C chromosomes and four different hybrids carrying 2n = 32 and 2n = 35 chromosomes. The genetic map lengths for each hybrid were established to compare their recombination rates. The rates were 1.4 and 2.7 times higher in the hybrids having C6 or C9 alone than in the control (0C). This enhancement reached 3.1 and 4.1 times in hybrids carrying six and nine C chromosomes, and it was also higher for each pair of hybrids carrying 2n = 32 or 2n = 35 chromosomes, with a dependence on which chromosomes remained as univalents. We have shown, for the first time, that the presence of one chromosome, C9 , affects significantly the recombination rate and reduces crossover interference. This result will have fundamental implications on the regulation of crossover frequency.

Shining fresh light on the evolution of photosynthesis

There are two types of photosynthesis, C3 and C4, and computational techniques have been used to explore how C4 plants evolved from their C3 ancestors.

Network function shapes network structure: the case of the Arabidopsis flower organ specification genetic network

The reconstruction of many biological networks has allowed detailed studies of their structural properties. Several features exhibited by these networks have been interpreted to be the result of evolutionary dynamics. For instance the degree distributions may follow from a preferential attachment of new genes to older ones during evolution. Here we argue that even in the absence of any evolutionary dynamics, the presence of atypical features may follow from the fact that the network implements certain functions. To examine this network function shapes network structure scenario, we focus on the Arabidopsis genetic network controlling early flower organogenesis in which gene expression dynamics has been modelled using a Boolean framework. Specifically, for a system with 15 master genes, the phenotype consists of 10 experimentally determined steady-state expression patterns, considered here as the functional constraints on the network. The space of genetic networks satisfying these constraints is sometimes referred to as the neutral or genotype network. We sample this space using Markov Chain Monte Carlo which allows us to demonstrate how the functional (phenotypic) constraints shape the gene network structure. We find that this shaping is strongest for the edge (interaction) usage, with effects that are functionally interpretable. In contrast, higher order features such as degree assortativity and network motifs are hardly shaped by the phenotypic constraints.

Intraspecific variation of recombination rate in maize

BACKGROUND

In sexually reproducing organisms, meiotic crossovers ensure the proper segregation of chromosomes and contribute to genetic diversity by shuffling allelic combinations. Such genetic reassortment is exploited in breeding to combine favorable alleles, and in genetic research to identify genetic factors underlying traits of interest via linkage or association-based approaches. Crossover numbers and distributions along chromosomes vary between species, but little is known about their intraspecies variation.

RESULTS

Here, we report on the variation of recombination rates between 22 European maize inbred lines that belong to the Dent and Flint gene pools. We genotype 23 doubled-haploid populations derived from crosses between these lines with a 50 k-SNP array and construct high-density genetic maps, showing good correspondence with the maize B73 genome sequence assembly. By aligning each genetic map to the B73 sequence, we obtain the recombination rates along chromosomes specific to each population. We identify significant differences in recombination rates at the genome-wide, chromosome, and intrachromosomal levels between populations, as well as significant variation for genome-wide recombination rates among maize lines. Crossover interference analysis using a two-pathway modeling framework reveals a negative association between re combination rate and interference strength.

CONCLUSIONS

To our knowledge, the present work provides the most comprehensive study on intraspecific variation of recombination rates and crossover interference strength in eukaryotes. Differences found in recombination rates will allow for selection of high or low recombining lines in crossing programs. Our methodology should pave the way for precise identification of genes controlling recombination rates in maize and other organisms.

Edge usage, motifs, and regulatory logic for cell cycling genetic networks

The cell cycle is a tightly controlled process, yet it shows marked differences across species. Which of its structural features follow solely from the ability to control gene expression? We tackle this question in silico by examining the ensemble of all regulatory networks which satisfy the constraint of producing a given sequence of gene expressions. We focus on three cell cycle profiles coming from baker's yeast, fission yeast, and mammals. First, we show that the networks in each of the ensembles use just a few interactions that are repeatedly reused as building blocks. Second, we find an enrichment in network motifs that is similar in the two yeast cell cycle systems investigated. These motifs do not have autonomous functions, yet they reveal a regulatory logic for cell cycling based on a feed-forward cascade of activating interactions.

A large maize (Zea mays L.) SNP genotyping array: development and germplasm genotyping, and genetic mapping to compare with the B73 reference genome

SNP genotyping arrays have been useful for many applications that require a large number of molecular markers such as high-density genetic mapping, genome-wide association studies (GWAS), and genomic selection. We report the establishment of a large maize SNP array and its use for diversity analysis and high density linkage mapping. The markers, taken from more than 800,000 SNPs, were selected to be preferentially located in genes and evenly distributed across the genome. The array was tested with a set of maize germplasm including North American and European inbred lines, parent/F1 combinations, and distantly related teosinte material. A total of 49,585 markers, including 33,417 within 17,520 different genes and 16,168 outside genes, were of good quality for genotyping, with an average failure rate of 4% and rates up to 8% in specific germplasm. To demonstrate this array's use in genetic mapping and for the independent validation of the B73 sequence assembly, two intermated maize recombinant inbred line populations - IBM (B73×Mo17) and LHRF (F2×F252) - were genotyped to establish two high density linkage maps with 20,913 and 14,524 markers respectively. 172 mapped markers were absent in the current B73 assembly and their placement can be used for future improvements of the B73 reference sequence. Colinearity of the genetic and physical maps was mostly conserved with some exceptions that suggest errors in the B73 assembly. Five major regions containing non-colinearities were identified on chromosomes 2, 3, 6, 7 and 9, and are supported by both independent genetic maps. Four additional non-colinear regions were found on the LHRF map only; they may be due to a lower density of IBM markers in those regions or to true structural rearrangements between lines. Given the array's high quality, it will be a valuable resource for maize genetics and many aspects of maize breeding.

Genome-wide crossover distribution in Arabidopsis thaliana meiosis reveals sex-specific patterns along chromosomes

In most species, crossovers (COs) are essential for the accurate segregation of homologous chromosomes at the first meiotic division. Their number and location are tightly regulated. Here, we report a detailed, genome-wide characterization of the rate and localization of COs in Arabidopsis thaliana, in male and female meiosis. We observed dramatic differences between male and female meiosis which included: (i) genetic map length; 575 cM versus 332 cM respectively; (ii) CO distribution patterns: male CO rates were very high at both ends of each chromosome, whereas female CO rates were very low; (iii) correlations between CO rates and various chromosome features: female CO rates correlated strongly and negatively with GC content and gene density but positively with transposable elements (TEs) density, whereas male CO rates correlated positively with the CpG ratio. However, except for CpG, the correlations could be explained by the unequal repartition of these sequences along the Arabidopsis chromosome. For both male and female meiosis, the number of COs per chromosome correlates with chromosome size expressed either in base pairs or as synaptonemal complex length. Finally, we show that interference modulates the CO distribution both in male and female meiosis.

Environmental versatility promotes modularity in genome-scale metabolic networks

BACKGROUND

The ubiquity of modules in biological networks may result from an evolutionary benefit of a modular organization. For instance, modularity may increase the rate of adaptive evolution, because modules can be easily combined into new arrangements that may benefit their carrier. Conversely, modularity may emerge as a by-product of some trait. We here ask whether this last scenario may play a role in genome-scale metabolic networks that need to sustain life in one or more chemical environments. For such networks, we define a network module as a maximal set of reactions that are fully coupled, i.e., whose fluxes can only vary in fixed proportions. This definition overcomes limitations of purely graph based analyses of metabolism by exploiting the functional links between reactions. We call a metabolic network viable in a given chemical environment if it can synthesize all of an organism's biomass compounds from nutrients in this environment. An organism's metabolism is highly versatile if it can sustain life in many different chemical environments. We here ask whether versatility affects the modularity of metabolic networks.

RESULTS

Using recently developed techniques to randomly sample large numbers of viable metabolic networks from a vast space of metabolic networks, we use flux balance analysis to study in silico metabolic networks that differ in their versatility. We find that highly versatile networks are also highly modular. They contain more modules and more reactions that are organized into modules. Most or all reactions in a module are associated with the same biochemical pathways. Modules that arise in highly versatile networks generally involve reactions that process nutrients or closely related chemicals. We also observe that the metabolism of E. coli is significantly more modular than even our most versatile networks.

CONCLUSIONS

Our work shows that modularity in metabolic networks can be a by-product of functional constraints, e.g., the need to sustain life in multiple environments. This organizational principle is insensitive to the environments we consider and to the number of reactions in a metabolic network. Because we observe this principle not just in one or few biological networks, but in large random samples of networks, we propose that it may be a generic principle of metabolic network organization.

Randomizing genome-scale metabolic networks

Networks coming from protein-protein interactions, transcriptional regulation, signaling, or metabolism may appear to have "unusual" properties. To quantify this, it is appropriate to randomize the network and test the hypothesis that the network is not statistically different from expected in a motivated ensemble. However, when dealing with metabolic networks, the randomization of the network using edge exchange generates fictitious reactions that are biochemically meaningless. Here we provide several natural ensembles of randomized metabolic networks. A first constraint is to use valid biochemical reactions. Further constraints correspond to imposing appropriate functional constraints. We explain how to perform these randomizations with the help of Markov Chain Monte Carlo (MCMC) and show that they allow one to approach the properties of biological metabolic networks. The implication of the present work is that the observed global structural properties of real metabolic networks are likely to be the consequence of simple biochemical and functional constraints.

Phenotypic robustness can increase phenotypic variability after nongenetic perturbations in gene regulatory circuits

Nongenetic perturbations, such as environmental change or developmental noise, can induce novel phenotypes. If an induced phenotype appears recurrently and confers a fitness advantage, selection may promote its genetic stabilization. Nongenetic perturbations can thus initiate evolutionary innovation. Genetic variation that is not usually phenotypically visible may play an important role in this process. Populations under stabilizing selection on a phenotype that is robust to mutations can accumulate such variation. After nongenetic perturbations, this variation can produce new phenotypes. We here study the relationship between a phenotype's mutational robustness and a population's potential to generate novel phenotypic variation. To this end, we use a well-studied model of transcriptional regulation circuits that are important in many evolutionary innovations. We find that phenotypic robustness promotes phenotypic variability in response to nongenetic perturbations, but not in response to mutation. Our work suggests that nongenetic perturbations may initiate innovation more frequently in mutationally robust gene expression traits.

CODA (crossover distribution analyzer): quantitative characterization of crossover position patterns along chromosomes

BACKGROUND

During meiosis, homologous chromosomes exchange segments via the formation of crossovers. This phenomenon is highly regulated; in particular, crossovers are distributed heterogeneously along the physical map and rarely arise in close proximity, a property referred to as "interference". Crossover positions form patterns that give clues about how crossovers are formed. In several organisms including yeast, tomato, Arabidopsis, and mouse, it is believed that crossovers form via at least two pathways, one interfering, the other not.

RESULTS

We have developed a software package--"CODA", for CrossOver Distribution Analyzer--which allows one to quantitatively characterize crossover patterns by fitting interference models to experimental data. Two families of interfering models are provided: the "gamma" model and the "beam-film" model. The user can specify single or two-pathways modeling, and the software package infers the model's parameters and their confidence intervals. CODA can handle data produced from measurements on bivalents or gametes, in the form of continuous crossover positions or marker genotyping. We illustrate the possibilities on data from Wheat, corn and mouse.

CONCLUSIONS

CODA extends the kind of crossover data that could be analyzed so far to include gametic data (rather than only bivalents/tetrads) when using two-pathways modeling. It will also enable users to perform analyses based on the beam-film model. CODA implements that model's complex physics and mathematics, and uses a summary statistic to overcomes the lack of a computable likelihood which has hampered its use till now.

Phenotypic plasticity can facilitate adaptive evolution in gene regulatory circuits

BACKGROUND

Many important evolutionary adaptations originate in the modification of gene regulatory circuits to produce new gene activity phenotypes. How do evolving populations sift through an astronomical number of circuits to find circuits with new adaptive phenotypes? The answer may often involve phenotypic plasticity. Phenotypic plasticity allows a genotype to produce different - alternative - phenotypes after non-genetic perturbations that include gene expression noise, environmental change, or epigenetic modification.

RESULTS

We here analyze a well-studied model of gene regulatory circuits. A circuit's genotype encodes the regulatory interactions among circuit genes, and its phenotype corresponds to a stable gene activity pattern the circuit forms. For this model, we study how genotypes are arranged in genotype space, where the distance between two genotypes reflects the number of regulatory mutations that set those genotypes apart. Specifically, we address whether this arrangement favors adaptive evolution mediated by plasticity. We find that plasticity facilitates the origin of genotypes that produce a new phenotype in response to non-genetic perturbations. We also find that selection can then stabilize the new phenotype genetically, allowing it to become a circuit's dominant gene expression phenotype. These are generic properties of the circuits we study here.

CONCLUSIONS

Taken together, our observations suggest that phenotypic plasticity frequently facilitates the evolution of novel beneficial gene activity patterns in gene regulatory circuits.

Quantifying slow evolutionary dynamics in RNA fitness landscapes

We re-examine the evolutionary dynamics of RNA secondary structures under directional selection towards an optimum RNA structure. We find that the punctuated equilibria lead to a very slow approach to the optimum, following on average an inverse power of the evolutionary time. In addition, our study of the trajectories shows that the out-of-equilibrium effects due to the evolutionary process are very weak. In particular, the distribution of genotypes is close to that arising during equilibrium stabilizing selection. As a consequence, the evolutionary dynamics leave almost no measurable out-of-equilibrium trace, only the transition genotypes (close to the border between different periods of stasis) have atypical mutational properties.

Distribution of essential interactions in model gene regulatory networks under mutation-selection balance

Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by few of the genes in the network. They also tend to have broad distributions for the out-degree. What mechanisms might be responsible for these degree distributions? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. There, we show that selection for a target expression pattern leads to the emergence of minimum connectivities compatible with the selective constraint. As a consequence, these gene networks have low in-degree and "functionality" is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by their essentiality. Furthermore, we find that mutations of the transcription factors drive the networks to have broad out-degrees. Finally, these classes of models are evolvable, i.e., significantly different genotypes can emerge gradually under mutation-selection balance.

Challenges in experimental data integration within genome-scale metabolic models

A report of the meeting "Challenges in experimental data integration within genome-scale metabolic models", Institut Henri Poincaré, Paris, October 10-11 2009, organized by the CNRS-MPG joint program in Systems Biology.

Return probabilities and hitting times of random walks on sparse Erdös-Rényi graphs

We consider random walks on random graphs, focusing on return probabilities and hitting times for sparse Erdös-Rényi graphs. Using the tree approach, which is expected to be exact in the large graph limit, we show how to solve for the distribution of these quantities and we find that these distributions exhibit a form of self-similarity.

Genotype networks in metabolic reaction spaces

Background

A metabolic genotype comprises all chemical reactions an organism can catalyze via enzymes encoded in its genome. A genotype is viable in a given environment if it is capable of producing all biomass components the organism needs to survive and reproduce. Previous work has focused on the properties of individual genotypes while little is known about how genome-scale metabolic networks with a given function can vary in their reaction content.

Results

We here characterize spaces of such genotypes. Specifically, we study metabolic genotypes whose phenotype is viability in minimal chemical environments that differ in their sole carbon sources. We show that regardless of the number of reactions in a metabolic genotype, the genotypes of a given phenotype typically form vast, connected, and unstructured sets -- genotype networks -- that nearly span the whole of genotype space. The robustness of metabolic phenotypes to random reaction removal in such spaces has a narrow distribution with a high mean. Different carbon sources differ in the number of metabolic genotypes in their genotype network; this number decreases as a genotype is required to be viable on increasing numbers of carbon sources, but much less than if metabolic reactions were used independently across different chemical environments.

Conclusions

Our work shows that phenotype-preserving genotype networks have generic organizational properties and that these properties are insensitive to the number of reactions in metabolic genotypes.

Two types of meiotic crossovers coexist in maize

We apply modeling approaches to investigate the distribution of late recombination nodules in maize (Zea mays). Such nodules indicate crossover positions along the synaptonemal complex. High-quality nodule data were analyzed using two different interference models: the "statistical" gamma model and the "mechanical" beam film model. For each chromosome, we exclude at a 98% significance level the hypothesis that a single pathway underlies the formation of all crossovers, pointing to the coexistence of two types of crossing-over in maize, as was previously demonstrated in other organisms. We estimate the proportion of crossovers coming from the noninterfering pathway to range from 6 to 23% depending on the chromosome, with a cell average of approximately 15%. The mean number of noninterfering crossovers per chromosome is significantly correlated with the length of the synaptonemal complex. We also quantify the intensity of interference. Finally, we develop inference tools that allow one to tackle, without much loss of power, complex crossover interference models such as the beam film. The lack of a likelihood function in such models had prevented their use for parameter estimation. This advance will allow more realistic mechanisms of crossover formation to be modeled in the future.

Effects of recombination on complex regulatory circuits

Mutation and recombination are the two main forces generating genetic variation. Most of this variation may be deleterious. Because recombination can reorganize entire genes and genetic circuits, it may have much greater consequences than point mutations. We here explore the effects of recombination on models of transcriptional regulation circuits that play important roles in embryonic development. We show that recombination has weaker deleterious effects on the expression phenotypes of these circuits than mutations. In addition, if a population of such circuits evolves under the influence of mutation and recombination, we find that three key properties emerge: (1) deleterious effects of mutations are reduced dramatically; (2) the diversity of genotypes in the population is greatly increased, a feature that may be important for phenotypic innovation; and (3) cis-regulatory complexes appear. These are combinations of regulatory interactions that influence the expression of one gene and that mitigate deleterious recombination effects.

Neutral network sizes of biological RNA molecules can be computed and are not atypically small

Background

Neutral networks or sets consist of all genotypes with a given phenotype. The size and structure of these sets has a strong influence on a biological system's robustness to mutations, and on its evolvability, the ability to produce phenotypic variation; in the few studied cases of molecular phenotypes, the larger this set, the greater both robustness and evolvability of phenotypes. Unfortunately, any one neutral set contains generally only a tiny fraction of genotype space. Thus, current methods cannot measure neutral set sizes accurately, except in the smallest genotype spaces.

Results

Here we introduce a generalized Monte Carlo approach that can measure neutral set sizes in larger spaces. We apply our method to the genotype-to-phenotype mapping of RNA molecules, and show that it can reliably measure neutral set sizes for molecules up to 100 bases. We also study neutral set sizes of RNA structures in a publicly available database of functional, noncoding RNAs up to a length of 50 bases. We find that these neutral sets are larger than the neutral sets in 99.99% of random phenotypes. Software to estimate neutral network sizes is available at .

Conclusion

The biological RNA structures we examined are more abundant than random structures. This indicates that their robustness and their ability to produce new phenotypic variants may also be high.

Adaptive networks of trading agents

Multiagent models have been used in many contexts to study generic collective behavior. Similarly, complex networks have become very popular because of the diversity of growth rules giving rise to scale-free behavior. Here we study adaptive networks where the agents trade "wealth" when they are linked together while links can appear and disappear according to the wealth of the corresponding agents; thus the agents influence the network dynamics and vice versa. Our framework generalizes a multiagent model of Bouchaud and Mézard [Physica A 282, 536 (2000)], and leads to a steady state with fluctuating connectivities. The system spontaneously self-organizes into a critical state where the wealth distribution has a fat tail and the network is scale free; in addition, network heterogeneities lead to enhanced wealth condensation.

Network of inherent structures in spin glasses: scaling and scale-free distributions

The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic "length scale" grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, for spin glasses based on random graphs, the degree distribution of the network of inherent structures exhibits a nontrivial scale-free tail.

New structural variation in evolutionary searches of RNA neutral networks

RNA secondary structure is an important computational model to understand how genetic variation maps into phenotypic (structural) variation. Evolutionary innovation in RNA structures is facilitated by neutral networks, large connected sets of RNA sequences that fold into the same structure. Our work extends and deepens previous studies on neutral networks. First, we show that even the 1-mutant neighborhood of a given sequence (genotype) G0 with structure (phenotype) P contains many structural variants that are not close to P. This holds for biological and generic RNA sequences alike. Second, we analyze the relation between new structures in the 1-neighborhoods of genotypes Gk that are only a moderate Hamming distance k away from G0, and the structure of G0 itself, both for biological and for generic RNA structures. Third, we analyze the relation between mutational robustness of a sequence and the distances of structural variants near this sequence. Our findings underscore the role of neutral networks in evolutionary innovation, and the role that high robustness can play in diminishing the potential for such innovation.