As a rule we take into account genetic regulation of biochemical pathways using analogous method to mathematical description of enzyme functioning. It also has same stages as such one for models of biochemical pathways. There is only one main difference. For mathematical description of genetic regulation we assign only two levels of specification – reduced and detail. In this material we show main steps of a development of the reduced description by example of transcription regulation.

At the first level of specification (reduced description) regulatory processes are taken into account with minimal number of parameters. For example, binding of transcription factors with operator or coeffector are defined using single summarized reaction without detail description of cooperativity effects or oligomeric structure of regulatory protein. In such case general scheme of repressor action can be represented as it’s shown on pic. 1 and inductor action as it’s shown on pic. 2. By this way we also can take into account various variants of catabolite transcription regulation, when activity of transcription factor is modulated against concentration of coeffector (as a rule, intermediate of regulating biochemical pathway).

Based on this scheme (pic. 1) total operator concentration (Op), repressor (Rep) and coeffector (Ef) can be represented by the following way:

Op = Op_free + Op_Rep;                       (Eq. 1) 

Rep = Rep_free + Ef_Rep + Op_Rep;    (Eq. 2) 

Ef = Ef_free + Ef_Rep;                           (Eq. 3) 

Here, Op_free, Rep_free and Ef_free are concentrations of free operator, repressor and coeffector, respectively; Op_Rep is a complex of repressor with operator and Ef_Rep – complex of repressor with effector.

Considering that as a rule concentration of transcription factor sufficiently higher than concentration of operator in the cell and by-turn sufficiently less than coeffector concentration equations 1 – 3 can be reduced by the following way.

Op = Op_free + Op_Rep;                (Eq. 4) 

Rep = Rep_free + Op_Rep;             (Eq. 5) 

Ef = Ef_free;                                    (Eq. 6) 

Concentration of repressor complexes with operator and coeffector can be redefined using dissociation constants of operator and coeffector respectively. So, equations (4-6) can be rewritten:

(Eq. 7,8)

In such case rate of transcription we can describe with help of equation (9), as though, when repressor attends in a system, transcription is possible only with free operator (pic. 1). It is caused that operator overlaps RNA polymerase (RNAP) binding site or lying before it directly interacts with transcription initiation factor (αCTD), wherefore blocking RNAP promotion on DNA chain [Schlax P.J., et al., 2003].

(Eq. 9)

where concentration of free operator can be defined from equations (7) and (8):

(Eq. 10)

Substituting equation (10) into (9), we can derive final equation of transcription rate (Eq. 11), when repressor attends in a system:

(Eq. 11)

Analogously we can derive transcription rate equation in case of induction although there is one difference. In such case transcription can be provided by two ways: only with help of inductor (pic. 2a) or jointly with transcription starting with free oprator (pic. 2b).

Pic. 2. General scheme of kinetic model describing induction. Ind – inductor; Kd_Ind – inductor binding constant to operator. All other designations are same as on pic. 1.

So, in case of the first variant (pic. 2a) transcription rate equation can be represented by following way:

(Eq. 12)

and the second one (pic. 2b):

(Eq. 13)

Using more detail specification level of genetic regulation mathematical description denominators of equations (11 - 13) contain more number of parameters, characterizing interactions between transcription factors and its operator or coeffector. Such way of description requires more amount of experimental information for model verification. If such information is available it is possible to use such kinetic models for specific studies of properties of transcription factor [Peskov K.V., et al., 2008].

1. Peskov K., Goryanin I., Prank K., Tobin F., Demin O. Kinetic Modeling of ace operon genetic regulation in Escherichia coli. J Bioinform Comput Biol (2008), 6(5): 933-959.
2. Demin O., Goryanin I. Kinetic Modelling in Systems Biology. Taylor & Francis (United States), (2008), pp.360

Moreover, at our website you can find materials describing results of application of this method to modeling of genetic regulatory elements.

1. Genetic regulation of arginine metabolism in Escherichia coli cells.
2. Kinetic model of ace operon genes expression regulation