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The aim of my communication is to show that known non-linear steps in transcription kinetics of multigene networks expression allow us to interpret such kinetics as new branch of synergetics. Transcription is defined as process of synthesis of a messenger RNA by RNA-polymerase. A template for polymerase is coding region of gene DNA. RNA-polymerase specifically binds to transcription start site and begins the synthesis. However, this process requires not only polymerase and template, but also assemblage of a multimolecular transcription machinery. The initiation of transcription is preceded by formation of preinitiation complex from general transcription factors and template DNA. Assemblage and activation of preinitiation complex is controlled by many protein factors (activators and repressors). Activators can function directly to regulate aspects of transcription preinitiation complex formation or function [1].
AUTOREGULATION are thought to provide an appealingly simple mechanism for maintaining and refining an expression pattern set up in response to a short-lived signal [2]. In the simplest case this involves the direct activation of a gene by its product. Autoregulation element contains, for example, Drosophila even-skipped and Deformed genes [3,4]. Alternatively, the gene product could stimulate expression indirectly either by enhancing the activity of activator or by reducing the level or activity of repressor. All three of these mechanisms result in a functionally equivalent situation - the enhancement and maintenance of an initially transient stimulus [5]. Indirect autoactivation by antirepression is assumed by [5] for Drosophila gene Kruppel.
COOPERATIVITY OR SYNERGISM is essential feature, conditioning non-linearity of transcription kinetics. Genes involved in regulation of development, as a rule, contain in their regulatory regions multiple identical (or similar) target sites. Multiple copies of regulatory elements act synergistically presumably due to some functional cooperativity of the transcription factors involved. Sharp activation threshold found to require multiple interacting targeting sites. The ability to activate transcription would depend upon the square or higher powers of transcription factors concentration [6].
COMPETITION. Two different regulatory elements can be closely linked or overlap, that causes the transcription factors to be competed for target sites [7]. The simplest mechanism by which DNA-binding transcriptional repressors can function to block gene expression is competitive DNA binding, where binding of the repressor prevents or interferes with binding of an activator.
HOMO- AND HETERODIMERIZATION occur among members of a family of transcription factors that share a common dimerization domain. The dimerization of factors generally brings about a change in sequence specificity or degree of activation. As the matter of fact, homo- and heterodimerization adds its own non-linear steps in transcription kinetics [8]. Amongst above mentioned genes, KRUPPEL protein, for example, is able to form a concentration-dependent homodimer and that KRUPPEL can activate transcription as a monomer, but losses this ability when forming homodimers at high concentrations of the protein [9].
DIFFUSION OF TRANSCRIPTION REGULATORS. We considering features of transcription kinetics in the case of unique gene element, that is in point case. In case of distributed system, apart from point DNA-protein interactions, we must take into consideration diffusion of transcription factors. Such situation is closely similar to that for Drosophila precleavaged blastula. Free diffusion for a small protein, e.g. a repressor or some other type of regulatory protein, is estimated at 1.5*10 cm sec , while for average protein diffusion coefficient is estimated at about 10 cm sec [10].
REACTION-DIFFUSION IN GENE NETS. Amongst many analyzed cases of gene-gene interactions in Drosophila segmentation gene network, next situation is common. Low concentration of X gene product activates expression of target gene Y, while high concentration of the X product represses the Y.
In early Drosophila embryo (precellularized blastula stage) the normal pair-rule pattern is of seven stripes of hairy gene expression, along main embryo axis, separated by intervening stripes of expressing runts gene. fushi-tarazu (ftz) and even-skipped (eve) show a similar mutually complementary pattern. The eve/ftz pattern is partially out of register with respect to the hairy/runt pattern. All pair-rule genes with exception of hairy and ftz eventually become expressed in 14 stripes (late pattern of expression).
In literature, we can find at least two models of trigger behavior of the pairs of genes with investigating structure. In accordance with [7] results there is cluster of some 20 targeting sites for HUNCHBACK protein binding in one of regulatory elements of even-skipped gene. The "multimerization" model of [8] suggests that hb protein activity is controlled by multimerization, with the protein being maximally active as a monomer. By this model hb is able to form multimers, either free in solution or when bound to DNA, and such forms of the protein are less efficient or nonfunctional as transcriptional activator. Accordingly to so called "titration" model, activation of the target gene is achieved only when some of the target sites are bound by the factor [11]. The promoter of the target gene contains multiple target sites and the specific factors might fail to activate the gene when either all of these sites are occupied or too few are occupied.
Edgar with coworkers [12] have been developing a model for gene-gene regulatory interactions based on mutual cooperative repression in such gene pair as hairy - runt. This system reminds many well-known examples of (bio)chemical kinetic triggers. Up-date interaction network of primary pair-rule genes hairy, runt, even-skipped plus secondary pair rule gene ftz reminds known examples of 4-reagent (two Xs and two Ys) RD systems (Cf. Fig.4 from [13] and schemes from [14]). In brief, we have following situation. Roles of two X morphogens can play RUNT and EVE proteins, because they mutually repress each other and eve gene is autoregulatory [3] and autoregulation is discussed for runt [15]. The function of second pair of Ys morphogens could play HAIRY and FTZ (and engrailed product). runt and hairy would act as potent repressors of each other and, in turn, HAIRY must act in the initiation of late eve expression in concert with eve autoregulation [15]. On the other hand, ftz regulated by eve similarly as in the case of late eve autoregulation and FTZ play negative regulatory role in generating of runt's late pattern. I execute computer simulation of this gene net activation both with stochastical model and with analytical one. In case of stochastical simulations the model describes changes in state of gene regulatory elements as consequence of binding/elimination, dimerization/dissociation and competition for targeting sites of site-specific transcription factors. In this case we simply have (in computer memory) string expressions that describe localization, strength of binding and mode of action of bound "factor" for each simulated gene regulatory region at each step of the calculation. Transcription/translation level (and, hence, probability of collision with given gene product) depends upon number and balance between bound activators and repressors.
In case of analytical simulations I use approach described in [16]. For example, it is believed that gene X is activated by product of gene Z (in accordance with "multimerization" model) and the X has self-enhancing element. A rate law comprising these features could be
,
The first member, with k > 1, describes auto-enhancing.
The member 
corresponds to a Hill type law which would yield cooperative activation
for small values of Z for n > 1. For m > n,
inhibition occurs for large values of Z. Cooperative activation
of the X gene by A-activator in case of competition between
A and repressor R we describe as follows [Cf.16]:
,
where A and R are concentrations of Activator and Repressor, correspondingly.
The processes in promoter elements are essentially non-linear and condition, as a result, Turing-like behavior of the system. Numerical experiments with these models reveal stable spatially periodic solutions and could help to explain and simulate of many known Drosophila mutations.
ZEBRA-STRIPPED
& RADIAL PATTERNS OF EXPRESSION OF GENE NETWORKS IN EMBRYO
At the same time, computer experiments demonstrate that, apart from outlined pair-rule gene-gene interactions, we must take into consideration both gap genes prepattern and global control by interplay of protein-kinase and protein-phosphatase cascades.
Tyrosine kinases adding phosphate groups to tyrosine residues of the target proteins, while tyrosine phosphatases play the opposite role, removing phosphate groups from tyrosine residues of the proteins. PTKs and PTPases are thought to play an equally important role in controlling cell differentiation. Final targets of the receptor tyrosine kinase cascades are nuclear specific transcription factors: their phosphorilation condition switching of the target genes expression [17,18]. Tyrosine phosphatases would guarantee that the activity of the tyrosine kinases would be transient, thus contributing to the desensibilization process. Initially one observes simple gradient pattern of Drosophila kinase D-raf activity [18]. But then, as I assume, appears and stabilizes 7-stripped pattern with stripes of high kinases activity and interstripes with low kinases and high phosphatases activity. Phosphorylation - dephosphorylation are essential for functions another segmentation gene products also. Soluble [cytoplasmic] kinases and phosphatases are low diffusing elements of the scheme, while small peptides -activators of the Receptor PTKs (torso) and Raf-like serine/theonine kinases, are relatovely fast diffusing components. Hence, zebra pattern of kinases/phosphatases activity could play a role of spatial framework for regulations and maintenance of stripe pattern of the pair-rule and another genes.
To my mind, one of essential examples of global Turing patterning functions of Drosophila PTK torso cascade are results of Warrior & Levine (1990) concerning consequences of eve patterning in torso gain-of-function (spliced) embryos. The gradual change in eve pattern in progressively more severe spliced embryos characterized gradual "gathering" of stripes in central part of embryo (from 56% of wt embryo length to 17% in severe spliced cases) in parallel with consequent loss of stripes: (8) - 7 - 6 - 4 - 1 (See Fig.5, [19]).
As A.Hunding discusses [16], theoreticians have pointed out that the "zebra" seven-stripped pattern may be generated by a truly symmetry breaking mechanism such as Turing's mechanism, that is, by autocatalytic reaction-diffusion system which is known to be capable of producing such stripes autonomously. The particular pair-rule stripes could then be activated by a combination of maternal, gap and Turing pattern interactions. One of the aims of my simulations is achievement of understanding in following question. Could we explain segmentation pattern dynamics on the baseground of interactions of known gap and pair-rule genes through autoregulation loops and cooperative activation/repression or we will have to introduce some sort of interactions of unknown morphogens governed by Turing mechanism? Now this question still remains open, but one of possibilities could be interplay between non-linear gene-gene interactions and PTK and PTPase cascades interactions.
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