An artificial neural network that utilizes hip joint actuations to control bifurcations and chaos in a passive dynamic bipedal walking model

Chaos is a central feature of human locomotion and has been suggested to be a window to the control mechanisms of locomotion. In this investigation, we explored how the principles of chaos can be used to control locomotion with a passive dynamic bipedal walking model that has a chaotic gait pattern. Our control scheme was based on the scientific evidence that slight perturbations to the unstable manifolds of points in a chaotic system will promote the transition to new stable behaviors embedded in the rich chaotic attractor. Here we demonstrate that hip joint actuations during the swing phase can provide such perturbations for the control of bifurcations and chaos in a locomotive pattern. Our simulations indicated that systematic alterations of the hip joint actuations resulted in rapid transitions to any stable locomotive pattern available in the chaotic locomotive attractor. Based on these insights, we further explored the benefits of having a chaotic gait with a biologically inspired artificial neural network (ANN) that employed this chaotic control scheme. Remarkably, the ANN was quite robust and capable of selecting a hip joint actuation that rapidly transitioned the passive dynamic bipedal model to a stable gait embedded in the chaotic attractor. Additionally, the ANN was capable of using hip joint actuations to accommodate unstable environments and to overcome unforeseen perturbations. Our simulations provide insight on the advantage of having a chaotic locomotive system and provide evidence as to how chaos can be used as an advantageous control scheme for the nervous system.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.     

Construction of an associative memory using unstable periodic orbits of a chaotic attractor

Unstable periodic orbits are the skeleton of a chaotic attractor. We constructed an associative memory based on the chaotic attractor of an artificial neural network, which associates input patterns to unstable periodic orbits. By processing an input, the system is driven out of the ground state to one of the pre-defined disjunctive areas of the attractor. Each of these areas is associated with a different unstable periodic orbit. We call an input pattern learned if the control mechanism keeps the system on the unstable periodic orbit during the response. Otherwise, the system relaxes back to the ground state on a chaotic trajectory. The major benefits of this memory device are its high capacity and low-energy consumption. In addition, new information can be simply added by linking a new input to a new unstable periodic orbit.     

Quantifying Missing Heritability at Known GWAS Loci

Recent work has shown that much of the missing heritability of complex traits can be resolved by estimates of heritability explained by all genotyped SNPs. However, it is currently unknown how much heritability is missing due to poor tagging or additional causal variants at known GWAS loci. Here, we use variance components to quantify the heritability explained by all SNPs at known GWAS loci in nine diseases from WTCCC1 and WTCCC2. After accounting for expectation, we observed all SNPs at known GWAS loci to explain more heritability than GWAS-associated SNPs on average (). For some diseases, this increase was individually significant: for Multiple Sclerosis (MS) () and for Crohn''s Disease (CD) (); all analyses of autoimmune diseases excluded the well-studied MHC region. Additionally, we found that GWAS loci from other related traits also explained significant heritability. The union of all autoimmune disease loci explained more MS heritability than known MS SNPs () and more CD heritability than known CD SNPs (), with an analogous increase for all autoimmune diseases analyzed. We also observed significant increases in an analysis of Rheumatoid Arthritis (RA) samples typed on ImmunoChip, with more heritability from all SNPs at GWAS loci () and more heritability from all autoimmune disease loci () compared to known RA SNPs (including those identified in this cohort). Our methods adjust for LD between SNPs, which can bias standard estimates of heritability from SNPs even if all causal variants are typed. By comparing adjusted estimates, we hypothesize that the genome-wide distribution of causal variants is enriched for low-frequency alleles, but that causal variants at known GWAS loci are skewed towards common alleles. These findings have important ramifications for fine-mapping study design and our understanding of complex disease architecture.     

Laboratory evolution of adenylyl cyclase independent learning in Drosophila and missing heritability

Gene interactions are acknowledged to be a likely source of missing heritability in large‐scale genetic studies of complex neurological phenotypes. However, involvement of rare variants, de novo mutations, genetic lesions that are not easily detected with commonly used methods and epigenetic factors also are possible explanations. We used a laboratory evolution study to investigate the modulatory effects of background genetic variation on the phenotypic effect size of a null mutation with known impact on olfactory learning. To accomplish this, we first established a population that contained variation at just 23 loci and used selection to evolve suppression of the learning defect seen with null mutations in the rutabaga adenylyl cyclase. We thus biased the system to favor relatively simplified outcomes by choosing a Mendelian trait and by restricting the genetic variation segregating in the population. This experimental design also assures that the causal effects are among the known 23 segregating loci. We observe a robust response to selection that requires the presence of the 23 variants. Analyses of the underlying genotypes showed that interactions between more than two loci are likely to be involved in explaining the selection response, with implications for the missing heritability problem.     

A revised indirect estimate of mutation rates in Amerindians

We have previously raised the possibility that the mutation rate resulting in rare electrophoretic variants is higher in tribal/tropical-dwelling/nonindustrialized societies than in civilized/temperate-dwelling/industrialized societies. Here, we report the results of examining 11 additional proteins for the occurrence of rare electrophoretic variants in 10 Amerindian tribes, for a total of 8,968 determinations and a total of 17,648 locus tests. When these data are combined with the results of all our previous similar studies of Amerindians, a total of 272,298 polypeptides, the products of 43 different loci, have been examined for the occurrence of rare electrophoretic variants. On the assumption that these variants are maintained by mutation pressure and are essentially neutral in their phenotypic effects, we have calculated by three different approaches that it requires an average mutation rate of 1.3 X 10(-5)/locus per generation to maintain the observed variant frequency. Concurrently, we are reporting elsewhere that a direct estimate of the mutation rate resulting in electromorphs in various studies of civilized industrialized populations is 0.3 X 10(-5)/locus per generation. Although this difference appears to have statistical significance, the nonquantifiable uncertainties to both approaches are such that our enthusiasm for a true difference in mutation rates between the two types of populations has diminished. However, even the lower of these estimates, when corrected for all the types of genetic variation that electrophoresis does not detect, implies total locus and gametic mutation rates well above those which in the past have dominated genetic thinking.     

Dynamical analysis of density-dependent selection in a discrete one-island migration model

A system of non-linear difference equations is used to model the effects of density-dependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior.     

Mining the LIPG allelic spectrum reveals the contribution of rare and common regulatory variants to HDL cholesterol

Genome-wide association studies (GWAS) have successfully identified loci associated with quantitative traits, such as blood lipids. Deep resequencing studies are being utilized to catalogue the allelic spectrum at GWAS loci. The goal of these studies is to identify causative variants and missing heritability, including heritability due to low frequency and rare alleles with large phenotypic impact. Whereas rare variant efforts have primarily focused on nonsynonymous coding variants, we hypothesized that noncoding variants in these loci are also functionally important. Using the HDL-C gene LIPG as an example, we explored the effect of regulatory variants identified through resequencing of subjects at HDL-C extremes on gene expression, protein levels, and phenotype. Resequencing a portion of the LIPG promoter and 5' UTR in human subjects with extreme HDL-C, we identified several rare variants in individuals from both extremes. Luciferase reporter assays were used to measure the effect of these rare variants on LIPG expression. Variants conferring opposing effects on gene expression were enriched in opposite extremes of the phenotypic distribution. Minor alleles of a common regulatory haplotype and noncoding GWAS SNPs were associated with reduced plasma levels of the LIPG gene product endothelial lipase (EL), consistent with its role in HDL-C catabolism. Additionally, we found that a common nonfunctional coding variant associated with HDL-C (rs2000813) is in linkage disequilibrium with a 5' UTR variant (rs34474737) that decreases LIPG promoter activity. We attribute the gene regulatory role of rs34474737 to the observed association of the coding variant with plasma EL levels and HDL-C. Taken together, the findings show that both rare and common noncoding regulatory variants are important contributors to the allelic spectrum in complex trait loci.     

The dynamics of density dependent population models

Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a strange attractor is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).Supported by NSF Grant No. BMS 74-21240.     

Bifurcations and chaos in a time-discrete integral model of population dynamics.

The properties of a time-discrete integral model of population dynamics are studied. The model equations define a dynamic system in the cone of nonnegative functions in L2. The presence of dissipator caused by spatial interaction is detected. Several bifurcations of codimensions 1 and 2 are investigated. The model may go over to chaotic behavior through a sequence of period-doubling bifurcations. The Lyapunov exponents and dimension of an attractor are found. The higher dimension of an attractor is shown to be a common feature of chaotic behavior.     

Machine learning on normalized protein sequences

Background

Directed protein evolution has been used to modify protein activity and research has been carried out to enhance the production of high quality mutant libraries. Many theoretical approaches suggest that allowing a population to undergo neutral selection may be valuable in directed evolution experiments.

Findings

Here we report on an investigation into the value of neutral selection in a classical model system for directed evolution, the conversion of the E. coli β-glucuronidase to a β-galactosidase activity. We find that neutral selection, i.e. selection for retaining glucuronidase activity, can efficiently identify the majority of sites of mutation that have been identified as beneficial for galactosidase activity in previous experiments. Each variant demonstrating increased galactosidase activity identified by our neutral drift experiments contained a mutation at one of four sites, T509, S557, N566 or W529. All of these sites have previously been identified using direct selection for beta galactosidase activity.

Conclusions

Our results are consistent with others that show that a neutral selection approach can be effective in selecting improved variants. However, we interpret our results to show that neutral selection is, in this case, not a more efficient approach than conventional directed evolution approaches. However, the neutral approach is likely to be beneficial when the resulting library can be screened for a range of related activities. More detailed statistical studies to resolve the apparent differences between this system and others are likely to be a fruitful avenue for future research.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Partitioning the Heritability of Tourette Syndrome and Obsessive Compulsive Disorder Reveals Differences in Genetic Architecture

The direct estimation of heritability from genome-wide common variant data as implemented in the program Genome-wide Complex Trait Analysis (GCTA) has provided a means to quantify heritability attributable to all interrogated variants. We have quantified the variance in liability to disease explained by all SNPs for two phenotypically-related neurobehavioral disorders, obsessive-compulsive disorder (OCD) and Tourette Syndrome (TS), using GCTA. Our analysis yielded a heritability point estimate of 0.58 (se = 0.09, p = 5.64e-12) for TS, and 0.37 (se = 0.07, p = 1.5e-07) for OCD. In addition, we conducted multiple genomic partitioning analyses to identify genomic elements that concentrate this heritability. We examined genomic architectures of TS and OCD by chromosome, MAF bin, and functional annotations. In addition, we assessed heritability for early onset and adult onset OCD. Among other notable results, we found that SNPs with a minor allele frequency of less than 5% accounted for 21% of the TS heritability and 0% of the OCD heritability. Additionally, we identified a significant contribution to TS and OCD heritability by variants significantly associated with gene expression in two regions of the brain (parietal cortex and cerebellum) for which we had available expression quantitative trait loci (eQTLs). Finally we analyzed the genetic correlation between TS and OCD, revealing a genetic correlation of 0.41 (se = 0.15, p = 0.002). These results are very close to previous heritability estimates for TS and OCD based on twin and family studies, suggesting that very little, if any, heritability is truly missing (i.e., unassayed) from TS and OCD GWAS studies of common variation. The results also indicate that there is some genetic overlap between these two phenotypically-related neuropsychiatric disorders, but suggest that the two disorders have distinct genetic architectures.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales ( e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Methodological Considerations in Estimation of Phenotype Heritability Using Genome-Wide SNP Data,Illustrated by an Analysis of the Heritability of Height in a Large Sample of African Ancestry Adults

Height has an extremely polygenic pattern of inheritance. Genome-wide association studies (GWAS) have revealed hundreds of common variants that are associated with human height at genome-wide levels of significance. However, only a small fraction of phenotypic variation can be explained by the aggregate of these common variants. In a large study of African-American men and women (n = 14,419), we genotyped and analyzed 966,578 autosomal SNPs across the entire genome using a linear mixed model variance components approach implemented in the program GCTA (Yang et al Nat Genet 2010), and estimated an additive heritability of 44.7% (se: 3.7%) for this phenotype in a sample of evidently unrelated individuals. While this estimated value is similar to that given by Yang et al in their analyses, we remain concerned about two related issues: (1) whether in the complete absence of hidden relatedness, variance components methods have adequate power to estimate heritability when a very large number of SNPs are used in the analysis; and (2) whether estimation of heritability may be biased, in real studies, by low levels of residual hidden relatedness. We addressed the first question in a semi-analytic fashion by directly simulating the distribution of the score statistic for a test of zero heritability with and without low levels of relatedness. The second question was addressed by a very careful comparison of the behavior of estimated heritability for both observed (self-reported) height and simulated phenotypes compared to imputation R² as a function of the number of SNPs used in the analysis. These simulations help to address the important question about whether today''s GWAS SNPs will remain useful for imputing causal variants that are discovered using very large sample sizes in future studies of height, or whether the causal variants themselves will need to be genotyped de novo in order to build a prediction model that ultimately captures a large fraction of the variability of height, and by implication other complex phenotypes. Our overall conclusions are that when study sizes are quite large (5,000 or so) the additive heritability estimate for height is not apparently biased upwards using the linear mixed model; however there is evidence in our simulation that a very large number of causal variants (many thousands) each with very small effect on phenotypic variance will need to be discovered to fill the gap between the heritability explained by known versus unknown causal variants. We conclude that today''s GWAS data will remain useful in the future for causal variant prediction, but that finding the causal variants that need to be predicted may be extremely laborious.     

Extranuclear differentiation and gene flow in the finite island model

Use of sequence information from extranuclear genomes to examine deme structure in natural populations has been hampered by lack of clear linkage between sequence relatedness and rates of mutation and migration among demes. Here, we approach this problem in two complementary ways. First, we develop a model of extranuclear genomes in a population divided into a finite number of demes. Sex-dependent migration, neutral mutation, unequal genetic contribution of separate sexes and random genetic drift in each deme are incorporated for generality. From this model, we derive the relationship between gene identity probabilities (between and within demes) and migration rate, mutation rate and effective deme size. Second, we show how within- and between-deme identity probabilities may be calculated from restriction maps of mitochondrial (mt) DNA. These results, when coupled with our results on gene flow and genetic differentiation, allow estimation of relative interdeme gene flow when deme sizes are constant and genetic variants are selectively neutral. We illustrate use of our results by reanalyzing published data on mtDNA in mouse populations from around the world and show that their geographic differentiation is consistent with an island model of deme structure.     

Fine mapping of five loci associated with low-density lipoprotein cholesterol detects variants that double the explained heritability

Complex trait genome-wide association studies (GWAS) provide an efficient strategy for evaluating large numbers of common variants in large numbers of individuals and for identifying trait-associated variants. Nevertheless, GWAS often leave much of the trait heritability unexplained. We hypothesized that some of this unexplained heritability might be due to common and rare variants that reside in GWAS identified loci but lack appropriate proxies in modern genotyping arrays. To assess this hypothesis, we re-examined 7 genes (APOE, APOC1, APOC2, SORT1, LDLR, APOB, and PCSK9) in 5 loci associated with low-density lipoprotein cholesterol (LDL-C) in multiple GWAS. For each gene, we first catalogued genetic variation by re-sequencing 256 Sardinian individuals with extreme LDL-C values. Next, we genotyped variants identified by us and by the 1000 Genomes Project (totaling 3,277 SNPs) in 5,524 volunteers. We found that in one locus (PCSK9) the GWAS signal could be explained by a previously described low-frequency variant and that in three loci (PCSK9, APOE, and LDLR) there were additional variants independently associated with LDL-C, including a novel and rare LDLR variant that seems specific to Sardinians. Overall, this more detailed assessment of SNP variation in these loci increased estimates of the heritability of LDL-C accounted for by these genes from 3.1% to 6.5%. All association signals and the heritability estimates were successfully confirmed in a sample of ～10,000 Finnish and Norwegian individuals. Our results thus suggest that focusing on variants accessible via GWAS can lead to clear underestimates of the trait heritability explained by a set of loci. Further, our results suggest that, as prelude to large-scale sequencing efforts, targeted re-sequencing efforts paired with large-scale genotyping will increase estimates of complex trait heritability explained by known loci.     

Complement component C3: molecular basis of the C3*S025 variant and evidence for molecular heterogeneity of other variants

Complement component 3 (C3) is the central molecule of the complement system. It displays a number of polymorphic variants with, as yet, unclear functional consequences. We have investigated a number of rare C3 variants by PCR-SSCP (polymerase chain reaction-single strand conformation polymorphism) analysis and could identify the molecular basis of a C3^*S025 variant. The decreased electrophoretic mobility of this protein is caused by the exchange of a neutral serine residue to an arginine residue (positively charged). This exchange is unlikely to have functional consequences as it maps to the C-terminus of the -chain. C3 variants appear to have originated from various independent mutations as we could not detect this mutation in different allotypes.