Forced evolution in silico by artificial transposons and their genetic operators: The ant navigation problem.

Modern evolutionary computation utilizes heuristic optimizations based upon concepts borrowed from the Darwinian theory of natural selection. Their demonstrated efficacy has reawakened an interest in other aspects of contemporary biology as an inspiration for new algorithms. However, amongst the many excellent candidates for study, contemporary models of biological macroevolution attract special attention. We believe that a vital direction in this field must be algorithms that model the activity of "genomic parasites", such as transposons, in biological evolution. Many evolutionary biologists posit that it is the co-evolution of populations with their genomic parasites that permits the high efficiency of evolutionary searches found in the living world. This publication is our first step in the direction of developing a minimal assortment of algorithms that simulate the role of genomic parasites. Specifically, we started in the domain of genetic algorithms (GA) and selected the Artificial Ant Problem as a test case. This navigation problem is widely known as a classical benchmark test and possesses a large body of literature. We add new objects to the standard toolkit of GA - artificial transposons and a collection of operators that operate on them. We define these artificial transposons as a fragment of an ant's code with properties that cause it to stand apart from the rest. The minimal set of operators for transposons is a transposon mutation operator, and a transposon reproduction operator that causes a transposon to multiply within the population of hosts. An analysis of the population dynamics of transposons within the course of ant evolution showed that transposons are involved in the processes of propagation and selection of blocks of ant navigation programs. During this time, the speed of evolutionary search increases significantly. We concluded that artificial transposons, analogous to real transposons, are truly capable of acting as intelligent mutators that adapt in response to an evolutionary problem in the course of co-evolution with their hosts.

Using evolutionary computations to understand the design and evolution of gene and cell regulatory networks.

This paper surveys modeling approaches for studying the evolution of gene regulatory networks (GRNs). Modeling of the design or 'wiring' of GRNs has become increasingly common in developmental and medical biology, as a means of quantifying gene-gene interactions, the response to perturbations, and the overall dynamic motifs of networks. Drawing from developments in GRN 'design' modeling, a number of groups are now using simulations to study how GRNs evolve, both for comparative genomics and to uncover general principles of evolutionary processes. Such work can generally be termed evolution in silico. Complementary to these biologically-focused approaches, a now well-established field of computer science is Evolutionary Computations (ECs), in which highly efficient optimization techniques are inspired from evolutionary principles. In surveying biological simulation approaches, we discuss the considerations that must be taken with respect to: (a) the precision and completeness of the data (e.g. are the simulations for very close matches to anatomical data, or are they for more general exploration of evolutionary principles); (b) the level of detail to model (we proceed from 'coarse-grained' evolution of simple gene-gene interactions to 'fine-grained' evolution at the DNA sequence level); (c) to what degree is it important to include the genome's cellular context; and (d) the efficiency of computation. With respect to the latter, we argue that developments in computer science EC offer the means to perform more complete simulation searches, and will lead to more comprehensive biological predictions.

Body plan evolvability: The role of variability in gene regulatory networks.

Recent computational modeling of early fruit fly (Drosophila) development has characterized the degree to which gene regulation networks can be robust to natural variability. In the first few hours of development, broad spatial gradients of maternally derived transcription factors activate embryonic gap genes. These gap patterns determine the subsequent segmented insect body plan through pair-rule gene expression. Gap genes are expressed with greater spatial precision than the maternal patterns. Computational modeling of the gap-gap regulatory interactions provides a mechanistic understanding for this robustness to maternal variability in wild-type (WT) patterning. A long-standing question in evolutionary biology has been how a system which is robust, such as the developmental program creating any particular species' body plan, is also evolvable, i.e. how can a system evolve or speciate, if the WT form is strongly buffered and protected? In the present work, we use the WT model to explore the breakdown of such Waddington-type 'canalization'. What levels of variability will push the system out of the WT form; are there particular pathways in the gene regulatory mechanism which are more susceptible to losing the WT form; and when robustness is lost, what types of forms are most likely to occur (i.e. what forms lie near the WT)? Manipulating maternal effects in several different pathways, we find a common gap 'peak-to-step' pattern transition in the loss of WT. We discuss these results in terms of the evolvability of insect segmentation, and in terms of experimental perturbations and mutations which could test the model predictions. We conclude by discussing the prospects for using continuum models of pattern dynamics to investigate a wider range of evo-devo problems.

The role of Bicoid cooperative binding in the patterning of sharp borders in Drosophila melanogaster.

In Drosophila embryonic development, the Bicoid (Bcd) protein establishes positional information of downstream developmental genes like hunchback (hb), which has a strong anterior expression and a sharp on-off boundary in the mid-embryo. The role of Bcd cooperative binding in the positioning of the Hb pattern has been previously demonstrated. However, there are discrepancies in the reported results about the role of this mechanism in the sharp Hb border. Here, we determined the Hill coefficient (nH) required for Bcd to generate the sharp border of Hb in wild-type (WT) embryos. We found that an n(H) of approximately 6.3 (s.d. 1.4) and 10.8 (s.d. 4.0) is required to account for Hb sharpness at early and late cycle 14A, respectively. Additional mechanisms are possibly required because the high nH is likely unachievable for Bcd binding to the hb promoter. To test this idea, we determined the nH required to pattern the Hb profile of 15 embryos expressing an hb14F allele that is defective in self-activation and found nH to be 3.0 (s.d. 1.0). This result indicates that in WT embryos, the hb self-activation is important for Hb sharpness. Corroborating our results, we also found a progressive increase in the required value of n(H) spanning from 4.0 to 9.2 by determining this coefficient from averaged profiles of eight temporal classes at cycle 14A (T1 to T8). Our results indicate that there is a transition in the mechanisms responsible for the sharp Hb border during cycle 14A: in early stages of this cycle, Bcd cooperative binding is primarily responsible for Hb sharpness; in late cycle 14A, hb self-activation becomes the dominant mechanism.

Evolution of the RNA world: From signals to codes.

The accumulated material in evolutionary biology, greatly enhanced by the achievements of modern synthetic biology, allows us to envision certain key hypothetical stages of prebiotic (chemical) evolution. This is often understood as the further evolution in the RNA World towards the RNA-protein World. It is a path towards the emergence of translation and the genetic code (I), signaling pathways with signaling molecules (II), and the appearance of RNA-based components of future gene regulatory networks (III). We believe that these evolutionary paths can be constructively viewed from the perspective of the concept of biological codes (Barbieri, 2003). Crucial evolutionary events in these directions would involve the emergence of RNA-based adaptors. Such adaptors connect two families of functionally and chemically distinct molecules into one functional entity. The emergence of primitive translation processes is undoubtedly the major milestone in the evolutionary path towards modern life. The key aspect here is the appearance of adaptors between amino acids and their cognate triplet codons. The initial steps are believed to involve the emergence of proto-transfer RNAs capable of self-aminoacylation. The second significant evolutionary breakthrough is the development of biochemical regulatory networks based on signaling molecules of the RNA World (ribonucleotides and their derivatives), as well as receptors and effectors (riboswitches) for these messengers. Some authors refer to this as the "lost language of the RNA World." The third evolutionary step is the emergence of signal sequences for ribozymes on the molecules of their RNA targets. This level of regulation in the RNA World is comparable to the gene regulatory networks of modern organisms. We believe that the signal sequences on target molecules have been rediscovered and developed by evolution into the gene regulatory networks of modern cells. In conclusion, the immense diversity of modern biological codes, in some of its key characteristics, can be traced back to the achievements of prebiotic evolution.

Problem of Domain/Building Block Preservation in the Evolution of Biological Macromolecules and Evolutionary Computation.

Structurally and functionally isolated domains in biological macromolecular evolution, both natural and artificial, are largely similar to "schemata", building blocks (BBs), in evolutionary computation (EC). The problem of preserving in subsequent evolutionary searches the already found domains / BBs is well known and quite relevant in biology as well as in EC. Both biology and EC are seeing parallel and independent development of several approaches to identifying and preserving previously identified domains / BBs. First, we notice the similarity of DNA shuffling methods in synthetic biology and multi-parent recombination algorithms in EC. Furthermore, approaches to computer identification of domains in proteins that are being developed in biology can be aligned with BB identification methods in EC. Finally, approaches to chimeric protein libraries optimization in biology can be compared to evolutionary search methods based on probabilistic models in EC. We propose to validate the prospects of mutual exchange of ideas and transfer of algorithms and approaches between evolutionary systems biology and EC in these three principal directions. A crucial aim of this transfer is the design of new advanced experimental techniques capable of solving more complex problems of in vitro evolution.

Canalization of gene expression in the Drosophila blastoderm by gap gene cross regulation.

Developing embryos exhibit a robust capability to reduce phenotypic variations that occur naturally or as a result of experimental manipulation. This reduction in variation occurs by an epigenetic mechanism called canalization, a phenomenon which has resisted understanding because of a lack of necessary molecular data and of appropriate gene regulation models. In recent years, quantitative gene expression data have become available for the segment determination process in the Drosophila blastoderm, revealing a specific instance of canalization. These data show that the variation of the zygotic segmentation gene expression patterns is markedly reduced compared to earlier levels by the time gastrulation begins, and this variation is significantly lower than the variation of the maternal protein gradient Bicoid. We used a predictive dynamical model of gene regulation to study the effect of Bicoid variation on the downstream gap genes. The model correctly predicts the reduced variation of the gap gene expression patterns and allows the characterization of the canalizing mechanism. We show that the canalization is the result of specific regulatory interactions among the zygotic gap genes. We demonstrate the validity of this explanation by showing that variation is increased in embryos mutant for two gap genes, Krüppel and knirps, disproving competing proposals that canalization is due to an undiscovered morphogen, or that it does not take place at all. In an accompanying article in PLoS Computational Biology (doi:10.1371/journal.pcbi.1000303), we show that cross regulation between the gap genes causes their expression to approach dynamical attractors, reducing initial variation and providing a robust output. These results demonstrate that the Bicoid gradient is not sufficient to produce gap gene borders having the low variance observed, and instead this low variance is generated by gap gene cross regulation. More generally, we show that the complex multigenic phenomenon of canalization can be understood at a quantitative and predictive level by the application of a precise dynamical model.

Gene expression noise in spatial patterning: hunchback promoter structure affects noise amplitude and distribution in Drosophila segmentation.

Positional information in developing embryos is specified by spatial gradients of transcriptional regulators. One of the classic systems for studying this is the activation of the hunchback (hb) gene in early fruit fly (Drosophila) segmentation by the maternally-derived gradient of the Bicoid (Bcd) protein. Gene regulation is subject to intrinsic noise which can produce variable expression. This variability must be constrained in the highly reproducible and coordinated events of development. We identify means by which noise is controlled during gene expression by characterizing the dependence of hb mRNA and protein output noise on hb promoter structure and transcriptional dynamics. We use a stochastic model of the hb promoter in which the number and strength of Bcd and Hb (self-regulatory) binding sites can be varied. Model parameters are fit to data from WT embryos, the self-regulation mutant hb(14F), and lacZ reporter constructs using different portions of the hb promoter. We have corroborated model noise predictions experimentally. The results indicate that WT (self-regulatory) Hb output noise is predominantly dependent on the transcription and translation dynamics of its own expression, rather than on Bcd fluctuations. The constructs and mutant, which lack self-regulation, indicate that the multiple Bcd binding sites in the hb promoter (and their strengths) also play a role in buffering noise. The model is robust to the variation in Bcd binding site number across a number of fly species. This study identifies particular ways in which promoter structure and regulatory dynamics reduce hb output noise. Insofar as many of these are common features of genes (e.g. multiple regulatory sites, cooperativity, self-feedback), the current results contribute to the general understanding of the reproducibility and determinacy of spatial patterning in early development.

In silico evolution of gene cooption in pattern-forming gene networks.

Gene recruitment or cooption occurs when a gene, which may be part of an existing gene regulatory network (GRN), comes under the control of a new regulatory system. Such re-arrangement of pre-existing networks is likely more common for increasing genomic complexity than the creation of new genes. Using evolutionary computations (EC), we investigate how cooption affects the evolvability, outgrowth and robustness of GRNs. We use a data-driven model of insect segmentation, for the fruit fly Drosophila, and evaluate fitness by robustness to maternal variability-a major constraint in biological development. We compare two mechanisms of gene cooption: a simpler one with gene Introduction and Withdrawal operators; and one in which GRN elements can be altered by transposon infection. Starting from a minimal 2-gene network, insufficient for fitting the Drosophila gene expression patterns, we find a general trend of coopting available genes into the GRN, in order to better fit the data. With the transposon mechanism, we find co-evolutionary oscillations between genes and their transposons. These oscillations may offer a new technique in EC for overcoming premature convergence. Finally, we comment on how a differential equations (in contrast to Boolean) approach is necessary for addressing realistic continuous variation in biochemical parameters.

Heuristic algorithms in evolutionary computation and modular organization of biological macromolecules: Applications to in vitro evolution.

Evolutionary computing (EC) is an area of computer sciences and applied mathematics covering heuristic optimization algorithms inspired by evolution in Nature. EC extensively study all the variety of methods which were originally based on the principles of selectionism. As a result, many new algorithms and approaches, significantly more efficient than classical selectionist schemes, were found. This is especially true for some families of special problems. There are strong arguments to believe that EC approaches are quite suitable for modeling and numerical analysis of those methods of synthetic biology and biotechnology that are known as in vitro evolution. Therefore, it is natural to expect that the new algorithms and approaches developed in EC can be effectively applied in experiments on the directed evolution of biological macromolecules. According to the John Holland's Schema theorem, the effective evolutionary search in genetic algorithms (GA) is provided by identifying short schemata of high fitness which in the further search recombine into the larger building blocks (BBs) with higher and higher fitness. The multimodularity of functional biological macromolecules and the preservation of already found modules in the evolutionary search have a clear analogy with the BBs in EC. It seems reasonable to try to transfer and introduce the methods of EC, preserving BBs and essentially accelerating the search, into experiments on in vitro evolution. We extend the key instrument of the Holland's theory, the Royal Roads fitness function, to problems of the in vitro evolution (Biological Royal Staircase, BioRS, functions). The specific version of BioRS developed in this publication arises from the realities of experimental evolutionary search for (DNA-) RNA-devices (aptazymes). Our numerical tests showed that for problems with the BioRS functions, simple heuristic algorithms, which turned out to be very effective for preserving BBs in GA, can be very effective in in vitro evolution approaches. We are convinced that such algorithms can be implemented in modern methods of in vitro evolution to achieve significant savings in time and resources and a significant increase in the efficiency of evolutionary search.

Modeling SELEX for regulatory regions using Royal Road and Royal Staircase fitness functions.

The field of evolutionary algorithms (EAs) emerged in the area of computer science due to transfer of ideas from biology and developed independently for several decades, enriched with techniques from probability theory, complexity theory and optimization methods. In this paper, we consider some recent results form the EAs theory transferred back into biology. The well-known biotechnological procedure SELEX (Systematic Evolution of Ligands by EXponential enrichment) is viewed as an experimental implementation of an evolutionary algorithm. Theoretical bounds on EAs runtime are applied to model SELEX search for a regulatory region consisting of promoter and enhancer sequences. A comparison of theoretical bounds to the results of computational simulation indicates some cases where the theoretical bounds give favorable prediction, while simulation requires prohibitive computational resource.

Quantitative Analysis of the Dynamics of Maternal Gradients in the Early Drosophila Embryo.

Predetermination, formation, and maintenance of the primary morphogenetic gradient (bicoid, bcd) of the early Drosophila embryo involves many interrelated processes. Here we focus on the biological systems analysis of the bcd mRNA redistribution in an early embryo. The results of the quantitative analysis of experimental data, together with the results of their dynamic modeling, substantiate the role of active transport in the redistribution of the bcd mRNA. The role of the nonlinearity of degradation mechanisms in the mRNA pattern robustness is discussed.

Quantification reveals early dynamics in Drosophila maternal gradients.

The Bicoid (Bcd) protein is a primary determinant of early anterior-posterior (AP) axis specification in Drosophila embryogenesis. This morphogen is spatially distributed in an anterior-high gradient, and affects particular AP cell fates in a concentration-dependent manner. The early distribution and dynamics of the bicoid (bcd) mRNA, the source for the Bcd protein gradient, is not well understood, leaving a number of open questions for how Bcd positional information develops and is regulated. Confocal microscope images of whole early embryos, stained for bcd mRNA or the Staufen (Stau) protein involved in its transport, were processed to extract quantitative AP intensity profiles at two depths (apical-under the embryo surface but above the nuclear layer; and basal-below the nuclei). Each profile was quantified by a two- (or three-) exponential equation. The parameters of these equations were used to analyze the early developmental dynamics of bcd. Analysis of 1D profiles was compared with 2D intensity surfaces from the same images. This approach reveals strong early changes in bcd and Stau, which appear to be coordinated. We can unambiguously discriminate three stages in early development using the exponential parameters: pre-blastoderm (1-9 cleavage cycle, cc), syncytial blastoderm (10-13 cc) and cellularization (from 14A cc). Key features which differ in this period are how fast the first exponential (anterior component) of the apical profile drops with distance and whether it is higher or lower than the basal first exponential. We can further discriminate early and late embryos within the pre-blastoderm stage, depending on how quickly the anterior exponential drops. This relates to the posterior-wards spread of bcd in the first hour of development. Both bcd and Stau show several redistributions in the head cytoplasm, quite probably related to nuclear activity: first shifting inwards towards the core plasm, forming either protrusions (early pre-blastoderm) or round aggregations (early nuclear cleavage cycles, cc, 13 and 14), then moving to the embryo surface and spreading posteriorly. These movements are seen both with the 2D surface study and the 1D profile analysis. The continued spreading of bcd can be tracked from the time of nuclear layer formation (later pre-blastoderm) to the later syncytial blastoderm stages by the progressive loss of steepness of the apical anterior exponential (for both bcd and Stau). Finally, at the beginning of cc14 (cellularization stage) we see a distinctive flip from the basal anterior gradient being higher to the apical gradient being higher (for both bcd and Stau). Quantitative analysis reveals substantial (and correlated) bcd and Stau redistributions during early development, supporting that the distribution and dynamics of bcd mRNA are key factors in the formation and maintenance of the Bcd protein morphogenetic gradient. This analysis reveals the complex and dynamic nature of bcd redistribution, particularly in the head cytoplasm. These resemble observations in oogenesis; their role and significance have yet to be clarified. The observed co-localization during redistribution of bcd and Stau may indicate the involvement of active transport.

Canalization of gene expression and domain shifts in the Drosophila blastoderm by dynamical attractors.

The variation in the expression patterns of the gap genes in the blastoderm of the fruit fly Drosophila melanogaster reduces over time as a result of cross regulation between these genes, a fact that we have demonstrated in an accompanying article in PLoS Biology (see Manu et al., doi:10.1371/journal.pbio.1000049). This biologically essential process is an example of the phenomenon known as canalization. It has been suggested that the developmental trajectory of a wild-type organism is inherently stable, and that canalization is a manifestation of this property. Although the role of gap genes in the canalization process was established by correctly predicting the response of the system to particular perturbations, the stability of the developmental trajectory remains to be investigated. For many years, it has been speculated that stability against perturbations during development can be described by dynamical systems having attracting sets that drive reductions of volume in phase space. In this paper, we show that both the reduction in variability of gap gene expression as well as shifts in the position of posterior gap gene domains are the result of the actions of attractors in the gap gene dynamical system. Two biologically distinct dynamical regions exist in the early embryo, separated by a bifurcation at 53% egg length. In the anterior region, reduction in variation occurs because of stability induced by point attractors, while in the posterior, the stability of the developmental trajectory arises from a one-dimensional attracting manifold. This manifold also controls a previously characterized anterior shift of posterior region gap domains. Our analysis shows that the complex phenomena of canalization and pattern formation in the Drosophila blastoderm can be understood in terms of the qualitative features of the dynamical system. The result confirms the idea that attractors are important for developmental stability and shows a richer variety of dynamical attractors in developmental systems than has been previously recognized.

Gene regulatory networks in Drosophila early embryonic development as a model for the study of the temporal identity of neuroblasts.

Genes belonging to the "gap" and "gap-like" family constitute the best-studied gene regulatory networks (GRNs) in Drosophila embryogenesis. Gap genes are a core of two subnetworks controlling embryonic segmentation: (hunchback, hb; Krüppel, Kr; giant, gt; and knirps, kni) and (hb; Kr; pou-domain, pdm; and, probably, castor, cas). Of particular interest is that (hb, Kr, pdm, cas) also specifies the temporal identity of stem cells, neuroblasts, in Drosophila neurogenesis. This GRN controls the sequential differentiation of neuroblasts during the asymmetric cell division. In the last decades, modeling of the patterning of gene ensemble (hb, Kr, gt, kni) in segmentation was in the center of attention. We show that our previously published and extensively studied model at a certain level of external factors is able to reproduce temporal patterns of (hb, Kr, pdm, cas) in neurogenesis with minor evolutionary explicable modifications. This result testifies in favor of a hypothesis that the similarity of two gene ensembles active in segmentation and neurogenesis is a result of co-option of the network architecture in evolution from the common ancestral form. By means of the model dynamical analysis, it is shown that the establishment of the robust patterns in both systems could be explained in terms of the action of attractors in the gap gene dynamical system. We formulate the common principles underlying the robustness of both GRNs in segmentation and neurogenesis due to the similar functional organization of the gene ensembles as having the same evolutionary origin.

Characterization of the Drosophila segment determination morphome.

Here we characterize the expression of the full system of genes which control the segmentation morphogenetic field of Drosophila at the protein level in one dimension. The data used for this characterization are quantitative with cellular resolution in space and about 6 min in time. We present the full quantitative profiles of all 14 segmentation genes which act before the onset of gastrulation. The expression patterns of these genes are first characterized in terms of their average or typical behavior. At this level, the expression of all of the genes has been integrated into a single atlas of gene expression in which the expression levels of all genes in each cell are specified. We show that expression domains do not arise synchronously, but rather each domain has its own specific dynamics of formation. Moreover, we show that the expression domains shift position in the direction of the cephalic furrow, such that domains in the anlage of the segmented germ band shift anteriorly while those in the presumptive head shift posteriorly. The expression atlas of integrated data is very close to the expression profiles of individual embryos during the latter part of the blastoderm stage. At earlier times gap gene domains show considerable variation in amplitude, and significant positional variability. Nevertheless, an average early gap domain is close to that of a median individual. In contrast, we show that there is a diversity of developmental trajectories among pair-rule genes at a variety of levels, including the order of domain formation and positional accuracy. We further show that this variation is dynamically reduced, or canalized, over time. As the first quantitatively characterized morphogenetic field, this system and its behavior constitute an extraordinarily rich set of materials for the study of canalization and embryonic regulation at the molecular level.

Spatial bistability generates hunchback expression sharpness in the Drosophila embryo.

During embryonic development, the positional information provided by concentration gradients of maternal factors directs pattern formation by providing spatially dependent cues for gene expression. In the fruit fly, Drosophila melanogaster, a classic example of this is the sharp on-off activation of the hunchback (hb) gene at midembryo, in response to local concentrations of the smooth anterior-posterior Bicoid (Bcd) gradient. The regulatory region for hb contains multiple binding sites for the Bcd protein as well as multiple binding sites for the Hb protein. Some previous studies have suggested that Bcd is sufficient for properly sharpened Hb expression, yet other evidence suggests a need for additional regulation. We experimentally quantified the dynamics of hb gene expression in flies that were wild-type, were mutant for hb self-regulation or Bcd binding, or contained an artificial promoter construct consisting of six Bcd and two Hb sites. In addition to these experiments, we developed a reaction-diffusion model of hb transcription, with Bcd cooperative binding and hb self-regulation, and used Zero Eigenvalue Analysis to look for multiple stationary states in the reaction network. Our model reproduces the hb developmental dynamics and correctly predicts the mutant patterns. Analysis of our model indicates that the Hb sharpness can be produced by spatial bistability, in which hb self-regulation produces two stable levels of expression. In the absence of self-regulation, the bistable behavior vanishes and Hb sharpness is disrupted. Bcd cooperative binding affects the position where bistability occurs but is not itself sufficient for a sharp Hb pattern. Our results show that the control of Hb sharpness and positioning, by hb self-regulation and Bcd cooperativity, respectively, are separate processes that can be altered independently. Our model, which matches the changes in Hb position and sharpness observed in different experiments, provides a theoretical framework for understanding the data and in particular indicates that spatial bistability can play a central role in threshold-dependent reading mechanisms of positional information.

Noise model estimation with application to gene expression.

Algorithms for the estimation of noise level and the detection of noise model are proposed. They are applied to gene expression data for Drosophila embryos. The 2D data on gene expression and the extracted 1D profiles are considered. Since the 1D data contain processing errors, an algorithm for separation of these processing errors is constructed to estimate the biological noise level. An approach to discrimination between the additive and multiplicative models is suggested for the 1D and 2D cases. Singular spectrum analysis and its 2D extension are exploited for the pattern extraction. The algorithms are tested on artificial data similar to the real data. Comparison of the results, which are obtained by the 1D and 2D methods, is performed for Krüppel and giant genes.

Relative sensitivity analysis of the predictive properties of sloppy models.

Commonly among the model parameters characterizing complex biological systems are those that do not significantly influence the quality of the fit to experimental data, so-called "sloppy" parameters. The sloppiness can be mathematically expressed through saturating response functions (Hill's, sigmoid) thereby embodying biological mechanisms responsible for the system robustness to external perturbations. However, if a sloppy model is used for the prediction of the system behavior at the altered input (e.g. knock out mutations, natural expression variability), it may demonstrate the poor predictive power due to the ambiguity in the parameter estimates. We introduce a method of the predictive power evaluation under the parameter estimation uncertainty, Relative Sensitivity Analysis. The prediction problem is addressed in the context of gene circuit models describing the dynamics of segmentation gene expression in Drosophila embryo. Gene regulation in these models is introduced by a saturating sigmoid function of the concentrations of the regulatory gene products. We show how our approach can be applied to characterize the essential difference between the sensitivity properties of robust and non-robust solutions and select among the existing solutions those providing the correct system behavior at any reasonable input. In general, the method allows to uncover the sources of incorrect predictions and proposes the way to overcome the estimation uncertainties.

Robustness of expression pattern formation due to dynamic equilibrium in gap gene system of an early Drosophila embryo.

The first manifestation of a segmentation pattern in the early Drosophila development is the formation of expression domains of genes belonging to the gap class. In our previous research the phenomenon of the gap system's robustness, exhibited as the ability to reduce highly variable gene expression in the course of development, was explained as a result of gene cross-regulation. In this paper we formulate the rigorous robustness conditions using the inherent properties of gap gene family. We propose an approach based on the observation that for the formation of a pattern with well-established domain borders it is necessary that there exist a stationary nucleus in which the gene expression level is almost constant in time. The dynamics of gap gene expression is described by a gene circuit model that correctly reproduces the observed principles of the border positioning. We take an advantage of the fact that the system behavior in a stationary nucleus is described by an algebraic equation and hence can be easily handled analytically. This enables us to explicitly characterize the gene cross-regulation properties guaranteeing the system robustness through the spatial behavior of the patterning system along the main embryo axis. In particular, it is proved that if the total regulatory action of all the genes acting in the border formation area changes synchronously with the initial gene gradient, the system will filtrate the initial positioning error and action of highly variable external input factors. We will now show how this approach can be applied as an instrument for the analysis of the robustness mechanism and revealing new biological findings.