Comment on comparative approach

J, theor. Biol. (1973) 42, 587-589

LETTERS TO THE EDITOR

Comment on Comparative

Approach

Recently, Finkel (1972~) proposed a pragmatic method for describing and predicting biological phenomena to circumvent the difficulties of the traditional reductionist approach. He stated that his comparative approach is consistent with quantum theory and suggested that his method of analogous systems does not suffer from the limitations of an atomistic approach. It will be shown here that it can simply be characterized as a method of interpolation and extrapolation and can be formulated without having to be couched in quantum mechanical terms. Except for the dubiously convenient resemblance in their mathematical forms, his final result does not seem to depend in any essential way on the effective Hamiltonian introduced. However, his restrictive final result, equation (I I), does depend cruciahy on further assumptions and approximations (Finkel, 1972b) to the quantum mechanical transition probabilities. Without these assumptions and approximations, his equation (lo), based upon which he arguesto arrive at the desired final result, does not obtain. It will be seen that, in our formulation, once the parametrized expression for the interested experimental quantity is written down, no further postulate or approximation is needed in order to obtain the final result. To prove this point, it suffices, without the loss of generality, to re-examine the example on the inhibition of specific antibody precipitation by various haptens considered by Finkel(1972a). The lowest two curves showing fractions precipitated vs. hapten concentrations illustrated in the reference are, as stated, obtained by interpolation between experimental points over the range of hapten concentrations considered. In order to make extrapolative predictions with respect to the chosen parameter 8 from the interpolated curves plotted as a function of concentration c, we choose to represent the fractions precipitated, as a first-order approximation, in the following form: f,(e,) = ~,+w4 (1) .fcw = 4+wh (2) for all values of c within the range of consideration. The parameters 8, and &, characterizing the relative positions of the haptens’ arsonate groups (Finkel, 1972a), are to be chosen for the purpose of extrapolation. Whether (1) and (2) are adequate representations, exact formal solutions to A, and B, always exist and are given by

4 = Big wm 2

1

587

e,f,(e,)],

588

A.

C.

CHEN

(4) Predictions

off,(e) are made by setting, on the basis of “analogousness”, f,(e) = A, + *, 0

(5)

and amount to an extrapolation with respect to the parameter 8, i.e. from Or = 2x/3 and 0, = rc to 0 = n/3 for any fixed value of c. Substitution of (3) and (4) into (5) yields

(6) which can be shown to be equivalent to equation (11) given in the reference. It is obvious that the first-order representations (1) and (2) leading to (5) and (6) may not always be adequate. However, we can easily generalize the above to the case where the quantity of interest f,(A), p and il being the two relevant parameters, needs to be represented by a Nth order polynomial in A for all values of p within the range of interest. Since biological systems are usually complex, the case where N becomes large may be potentially important. To generalize to an arbitrary order iV, we write ./p(A)

=

f

n=O

(Ap)*127

(7)

for i = I, 2, . . . , N+ 1 with f&4

= 2 (Ap)n~ n=O

(8)

for extrapolative or interpolative prediction purposes as the case may be. (7) represents a set of N+ 1 inhomogeneous linear equations for the N+ 1 expansion coefficients (A&. The solution to these equations may easily be obtained from Cramer’s rule. With these coefficients determined, (8) can be written in the compact form (9) It is perhaps of interest to point out that equation (9) has the same form as the well-known interpolation formula of Lagrange, which has been derived in the present context for prediction purposes. It is easily recognized that the usefulness of the method depends upon the availability of the experimental dataf,&) for the Nf 1 “reference” systems.

LETTERS

TO

THE

589

EDITOR

For N = 2, we have from (9) A2 & - A(& -I- A,) + A2 & I, --I(& + A,) + A2 MA) = (A, -,q,)(j& -2,) fp@J + (A, -A.,)@, --A,) -f&) + “i(:: $;:y2fp(&, 3

1

3

2

wheW&l)&(~2) andfp(~3) re Present the needed experimental information. When N becomes large, the needed information may be difficult to come by. Under such circumstances, the proposed method of “analogous” systems may indeed cease to be pragmatic. We believe that, in terms of our exact formulation and straightforward characterization, the basic idea of the proposed comparative approach as well as its limitations has been made explicit. Physics Department, S. John’s University, Jamaica, N. Y. 11439, U.S.A. (Received 16 February 1973, and in revisedform 7 May 1973)

FINKEL, FINKEL,

REFERENCES (1972a). J. theor. Biol. 37, 245. W. (19726). Am. J. Phys. 40, 1028.

R. W. R.

AUGUSTINE C.

CHEN