Application of the models

By means of mathematical models, it is possible to produce the following operations:

1) Numerical modeling of dynamical characteristics of biological systems under very different conditions acting on them, such as:

• viewing the reaction of a gene network in response to alteration of initial concentration values of the system’s components;

• viewing the reaction of a gene network in response to alteration of the values of concentrations of the system’s components under the fixed periods of time;

• viewing the gene network behavior in response to alteration of parameters determining the rates of elementary processes;

• computer simulation of gene mutation (disruption of gene or enzyme function, alteration of rates of translation, transcription, transportation, utilization processes etc.) by varying the parameters of a gene network mathematical model;

2) Comparative analysis of theoretical numerical data to the real experimental ones.

Mathematical model of the gene network on cholesterol biosynthesis regulation

Below, the examples are given that briefly describe several real mutations as well as the parameters, modification of which in mathematical model may simulate the mutations of interest.

Computer simulation of mutations of the Low Density Lipoprotein (LDL) receptor gene

The lack of LDL receptors leads to considerable increase of concentration of these particles in the blood. Exclusively high LDL cholesterol level in blood is observed in the homozygous form of hypercholesterinemia, when the patient is deficient by LDL receptors [Whiting and Elliott, 1972]. High level of LDL circulating in blood throughout 4,5 days in heterozygotes and 6 days in homozygotes (instead of 2,5 days in the norm) may facilitate the process of their modification (chemical reactions, e.g., peroxide oxidation, decialization, glucosylation, and others). These alterations are important for development of atherosclerosis, because exactly modified but not native LDL are characterized by the atherogenic properties. The reticuloendothelial cells more easily pick up the modified LDL rather than the unmodified one. The macrophages that actively ingest modified LDL are being transformed to the foam cells. The later, being overwhelmed by the cholesterol particles, are dying. This process leads to the storing of cholesterol in the artery intima and to formation, first, of the lipid spots (bars) and then of the atherosclerotic plaques.

A variety of mutations in the LDL receptor gene can be subdivided into 5 main classes (according to their influence on receptor functioning).

I-st class mutations:

This class is represented by mutations causing production of the protein, which is not detected by immune system, or the so-called null-allele mutations. Many null-allele mutations cause decrease of LDL receptor mRNA concentration in the cells of patients. Moreover, in some patients, the relevant mRNA is not produced at all.

Recommended constant values:

K_i = 1 sec^-1; KISBldlr

K_m = 10³ molecules/cell. KMSBldlr

By varying these constant values, it is possible to simulate the 1st class mutations.

The 2^nd class mutations cause delay in transport of receptor protein from the EPR (the place of its synthesis) to the cell membrane. This is produced by disruption of transformation of immature protein with M = 120 kD into its mature form with M = 160 kD. The defective protein has improper polypeptide chain spatial organization. As a consequence, the protein can not transport to the Golgi apparatus in processing and it is degraded in the EPR [Esser and Russel, 1988].

LDLRm dislocation from the cell volume to the cell surface:

Recommended constant value: K = 10^-2 sec^-1. Kfre____

By varying these constant values, it is possible to simulate the 2nd class mutations.

Under the 3^rd class mutations, the synthesis of receptor and its transport to the cell surface are normal. However, the receptor may poorly bind a ligand (LDL), although in most cases it preserves an ability to interact with the b -VLDL (lipoprotein of very low density). This mutation is often produced by deletion, which removes 1^st and 2^nd repeats from the ligand-binding domain of the LDL receptor gene. Deletion of a single aminoacid in the conserved triplet of the cystein-rich repeat (Ser-Asp-Glu) produces the similar effect as deletion of the whole repeat [Russel et al., 1989].

In order to provide better attachment of receptor with the LDL protein, not only the ligand-binding domain should function properly, but also the neighboring domain influencing the spatial orientation of the first domain. If insertion is located in the second domain, an ability of receptor to bind LDLP also falls.

Recommended constant values:

K₁ = 10^-8 (molecules/cell)^-1c^-1 Kclaw1

K₂ = 10^-4 sec^-1. Kclaw2

By varying these constant values, it is possible to simulate the 3rd class mutations.

The 4^th class mutations are characterized by the synthesis of defective receptor, which is unable to form the clasters in the clathrin coated pits. In its turn, this event prevents internalization of the LDL attached to receptor into the cell. In particular, among this group of mutations are those mutant alleles, which govern the synthesis of restricted receptors lacking the cytoplasmic and trans-membrane domains. This prevents the receptor to keep at the cell surface and favors to its secretion out of the cell limits.

Recommended constant value: K = 1.93*10^-6 sec^-1. KdegLDLR

Recommended constant value: K = 1.93*10^-6 sec^-1. KdegLRLo

By varying these constant values, it is possible to simulate the 4^th class mutations.

The cycle of receptor modification in a cell ends by its dissociation from the ligand in the endosomes at acidic pH. Then receptor returns to the cell surface. In case of the 5^th class mutations, LDL receptor protein is shorter in length. Due to this reasoning, the protein is unable to dissociate from the ligand in endosomes, thus, the receptor degrades. Mutations of this class are primarily localized in the second domain, which is homologous to precursor of the epidermal growth factor. The rate of receptor’s degradation by the 5^th class mutations determined in the local population of South Africa may be increased 5-10 fold, thus, decreasing essentially the number of receptors at the cell surface.

Cholesterol release from LDL in the cell:

Recommended constant value: K = 10^-2 sec^-1. KISBfree

Recommended constant value: K = 1.93*10^-6 sec^-1. KdegLRLi

By varying these constant values, it is possible to simulate the 5^th class mutations.

Mathematical model of the gene network on erythroid cell differentiation regulation

In what follows, the examples are given in which several particular mutations are briefly described and parameters are indicated, alterations of which in mathematical model may simulate the mutations considered.

Computer simulation of mutations of the GATA-1 gene:

The transcription factor GATA-1 is a key regulator of erythroid-cell differentiation and survival. The transcriptional cofactor CREB-binding protein (CBP) binds to the zinc finger domain of GATA-1, markedly stimulates the transcriptional activity of GATA-1, and is required for erythroid differentiation. CBP, but not p/CAF, acetylates GATA-1 at two highly conserved lysine-rich motifs present at the C-terminal tails of both zinc fingers. Using [3H]acetate labelling experiments and anti-acetyl lysine immunoprecipitations, GATA-1 is acetylated in vivo at the same sites acetylated by CBP in vitro. In addition, CBP stimulates GATA-1 acetylation in vivo in an E1A-sensitive manner, thus establishing a correlation between acetylation and transcriptional activity of GATA-1. Acetylation in vitro did not alter the ability of GATA-1 to bind DNA, and mutations in either motif did not affect DNA binding of GATA-1 expressed in mammalian cells. Since certain functions of GATA-1 are revealed only in an erythroid environment, GATA-1 constructs were examined for their ability to trigger terminal differentiation when introduced into a GATA-1-deficient erythroid cell line. Mutations in either acetylation motif partially impaired the ability of GATA-1 to induce differentiation while mutations in both motifs abrogated it completely. Taken together, these data indicate that CBP is an important cofactor for GATA-1 and suggest a novel mechanism in which acetylation by CBP regulates GATA-1 activity in erythroid cells [Hung et al., 1999].

Haematopoietic development is regulated by nuclear protein complexes that coordinate lineage-specific patterns of gene expression. Targeted mutagenesis in embryonic stem cells and mice has revealed roles for the X-linked gene GATA-1 in erythrocyte differentiation. GATA-1 is the founding member of a family of DNA-binding proteins that recognize the motif WGATAR through a conserved multifunctional domain consisting of two C4-type zinc fingers. A family with X-linked dyserythropoietic anaemia due to a substitution of methionine for valine at amino acid 205 of GATA-1. This highly conserved valine is necessary for interaction of the amino-terminal zinc finger of GATA-1 with its essential cofactor, FOG-1 (for friend of GATA-1). The V205M mutation abrogates the interaction between GATA-1 and Fog-1, inhibiting the ability of GATA-1 to rescue erythroid differentiation in an erythroid cell line deficient for GATA-1 (G1E). FOG-1:GATA-1 associations are very importan in erythroid development. It was suggested that other X-linked anaemias may be caused by defects in GATA-1. [Nichols et al., 2000]

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmrg

K_m = 10² molecules/cell KMSBgmrg

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmre

K_m = 10² molecules/cell KMSBgmre

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmas

K_m = 10² molecules/cell KMSBgmas

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmad

K_m = 10² molecules/cell KMSBgmad

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmpb

K_m = 10² molecules/cell KMSBgmpb

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmrc

K_m = 10² molecules/cell KMSBgmrc

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmrf

K_m = 10² molecules/cell KMSBgmrf

Recommended constant values:

K_i = 3.54*10^-1 sec^-1 KISBgmag

K_m = 10² molecules/cell KMSBgmag

Recommended constant values:

K_i = 1.54*10^-1 sec^-1 KISBgmbg

K_m = 10² molecules/cell KMSBgmbg

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmrt

K_m = 10² molecules/cell KMSBgmrt

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmrh

K_m = 10² molecules/cell KMSBgmrh

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmre

K_m = 10² molecules/cell KMSBgmre

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmtr

K_m = 10² molecules/cell KMSBgmtr.

By varying these constant values, one may model mutations of the GATA-1 gene.

Computer simulation of mutations of the EKLF gene:

The erythroid Kruppel-like factor (EKLF) is a key regulatory protein in globin gene expression. This zinc finger transcription factor is required for expression of the adult beta globin gene, and it has been suggested that it plays an important role in the developmental switch from fetal gamma to adult beta globin gene expression. A sequence element in the distal promoter region of the mouse EKLF gene is critical for the expression of this transcription factor. The element consists of an E box motif flanked by 2 GATA-1 binding sites. The mutation of the E box or the GATA-1 consensus sequences eliminates expression from the EKLF promoter in transgenic mice. These results confirm the importance of this activator element for in vivo expression of the EKLF gene [Anderson et al., 2000].

Recommended constant values:

K_i = 10^-1 sec^-1 KISBgmre

K_m = 10² molecules/cell KMSBgmre

By varying constant values given above, one may model mutations of the EKLF gene.

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Esser V., Russel D. W. Transport-deficient mutations in the low density lipoprotein receptor. Alternation in the cystein-rich and cystein-poor regions of the protein block intracellular transport. The Journal of Biological Chemistry, 1988,  V. 263,  P. 13276 – 13281.
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Russel D. W., Esser V., Hobbs H. H. Molecular basis of familiar hypercholesterolemia. Ateriosclerosis. Suppl., 1989, V. 9, P. 8 – 13.
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Hung HL; Lau J; Kim AY; Weiss MJ; Blobel GA CREB-Binding protein acetylates hematopoietic transcription factor GATA-1 at functionally important sites. Mol Cell Biol, 1999 May, 19:5, 3496-3505.
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Nichols KE, Crispino JD, Poncz M, White JG, Orkin SH, Maris JM, Weiss MJ Familial dyserythropoietic anaemia and thrombocytopenia due to an inherited mutation in GATA1. Nat Genet 2000 Mar;24(3):266-270.
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