Tachyphylaxis and Tolerance: Biomathematics of Rate Sensitivity

Tachy 1Fig.  1

Tachyphylaxis is indicated by the transient responses to four constant excitations at various levels but for the same duration. The spiky responses all return to the same basal (steady state) level. By the end of the excitation negative responses occur, i.e.,below the basal level after the sudden discontinuation of the constant excitation.

 

Fig. 2

Often repeated short excitation result in diminished responses, as in the drug tolerance development, here shown by the two consecutive short excitation of the same intensity. This lack of system responsiveness to short repeated excitations can be compensated by increasing the intensity of further short excitation, which in the practical application may result in an addictive condition. Furthermore, if enough time is allowed to elapse between the two consecutive same excitations (compare the first and the second excitation at the end of the graph), the response nearly returns to the height of response of the first excitation.

 Tachy 2

Tachy3Fig. 3

Various ramp excitations show the typical feature of the rate sensitivity, namely, the faster the ramp, the higher is the transient response. With a slow enough ramp (to the same level of the final excitation) the response may disappear altogether.

MODEL description

The response of a biological system to the external excitation (stimuli) is generally expected to be proportional (not in mathematical sense), i.e., the sustained excitation would produce a similarly sustained response. There are, however, cases of transient response which do not follow that expectation, such as in the phenomena of tachyphylaxis and tolerance. In tachyphylaxis (ref A) there is a sudden decrease in the response after a drug administration in spite of maintenance of the drug level. When a subject’s reaction show reduction of the drug effect after a repeated use of the drug, this is called a drug tolerance development (ref B).

In physical systems the term of parametric excitation was coined for the system excitation by variations in parameters (ref C). These system were not driven by directly affecting the system variables, but the system response is generated by the changes in the parameters.

In biological systems of molecular interactions the system response can also vary by not introducing an activating factor as a part of the biological molecular species, but by modifying the enzyme (receptor) activity through parametric excitation as shown in the diagram

R —–> A –p(S(t))–> B —q—>

Here R stands for the rate of production A, the inactive enzyme and B stands for the active enzyme. The parameter p(S(t)) is expressed in terms of the excitation intensity S as a function of time t, the relaxation parameter q is a rate constant.

This dynamical system is the simplest open system, represented by two differential equations

dA/dt = R – p(S(t))A

dB/dt = p(S(t))A – qB

which in the steady state dictate B = R/q at any excitation level. For a constant excitation S the transient state B(t) yields (analytically) a two-exponential spiky response as shown in Fig. 1 . However, for a variable excitation S(t), the system is not always analytically solvable and a numerical such as Runge-Kutta method (ref D) is required for a numerical solution. In what follows, this method is used to produce the graphical images shown in Fig. 1 through Fig. 3.

When sufficiently complete data on the excitation and response patterns are available, Marquardt least squares (ref D) may allow to estimate the type of molecular interactions similar to those discussed here in the simplest model.

The figures 1 through 3 show in graphical representation that the single simple parametric excitation model can account for the majority of effects on system response by examining it with several stimulatory patterns. The system diagrams and mathematical description can be achieved by the least squares method and suggest specifics of molecular interactions involved in the production of responses. The structure of the system and the values of the parameters may allow a quantitative comparison of the variety of the experimental conditions.

REFERENCES

A. Tachyphylaxis, Wikipedia

B. Tolerance, Wikipedia

C. Minorski N., Journal of the Franklin Institute, Volume 240, Issue 1, July 1945, pp.25-46

D. Press W H, Flannery B P, Teukolsky SA, Vetterling W T: Numerical Recipes in C; Cambridge University Press, Cambridge, 1988

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