Analysis of Antibody by Real-Valued Special Functions

Antibodies have been proven to be indispensable tools for biomedical applications. Different engineered antibodies have been developed for various purposes according to the amino acid sequence and/or spatial structure of protein (Figure 1). At present, it is still difficult to predict the optimal structure of antibodies. Topology knowledge can be important in antibody application as well as transformation. Theoretically, we can obtain desired antibodies by using protein/gene engineering technology. For instance, we can transform the Complementarity Determining Region (CDR) to promote the affinity of the antibody to antigen. Similarly, we could also transform any domain of antibody to make it bind with any desired target. Under this vision, topology is a powerful tool to predict the structure of protein and it will serve to antibody engineering. Our present work tries to explain, and predict, if possible, the change of structure, size and function of antibodies as well as their fragments from a topological perspective.

 

For the classical Euler’s gamma function and psi (digamma) function are defined by respectively.



The derivatives for are known as polygamma functions. For (see [1]), the following series representations are established:







where denotes the Euler’s constant.

We next recall [2-5] that a function is said to be completely monotonic on an interval , if has derivatives of all orders on which alternate successively in sign, that is,



for all and for all . If inequality (1.5) is strict for all and all , then is said to be strictly completely monotonic.

The classical Bernstein–Widder theorem [6, p. 160, Theorem 12a] states that a function is completely monotonic on if and only if it is a Laplace transform of some nonnegative measure , that is,



where is non-decreasing and the integral converges for .

We recall also [7-9] that a positive function is said to be logarithmically completely monotonic on an interval if has derivatives of all orders on and



for all and for all . If inequality (1.7) is strict for all and all , then is said to be strictly logarithmically completely monotonic.

The antibody structure will be changed when it binds certain targets (Figure 2a), i.e., antigen, receptor. How to describe the changes in the view of topology? The following cases will explain it in detail.

 

It was proved explicitly in [8] and other articles that a logarithmically completely monotonic function must be completely monotonic. 

In [10], G. D. Anderson et al. proved that the function



is strictly decreasing and strictly convex on , with two limits



From (1.9) and the monotonicity of , then the double inequalities



holds for all .

In [11, Theorem 1], by using the well-known Binet's formula, H. Alzer generalized the monotonicity and convexity of , that is, the function is strictly completely monotonic on if and only if .



In [12], D. Kershaw and A. Laforgia proved that the function is decreasing on and is increasing on . These are equivalent to the function being increasing and being decreasing on , respectively.

In [13, Theorem 5], F. Qi and Ch.-p. Chen generalized these functions. They obtained the fact that for all the function is strictly increasing for and strictly decreasing for , respectively.

After the papain digestion, the remained antibody functional part (usually the Fab domain), will be smaller and the structure is also changed (Figure 1b). These changes can be revealed vividly using topology. Recently [14, Theorem 1], F. Qi, C.-F Wei and B.-N Guo established another excellent result, which states that forgiven and , let



The function (1.12) is logarithmically completely monotonic with respect to if and only if ; and if , the reciprocal of the function (1.12) is logarithmically completely monotonic with respect to .

Antibodies occur spontaneously gathering and forming dimer, polymer, which will influence their functions (Figure 2b). In antibody engineering practice, it urgently needs some measures to overcome this difficulty. From topology perspective, we could understand this issue as follow.

Stimulated by the above results, we put forward the function as follows: forgiven and real number , let the function be defined by our first result is contained in the following theorem.

For the function (1.13), then the following statements are true:

(1) for any given , the function (1.13) is strictly logarithmically completely monotonic with respect to if and only if ;

(2) for any given , if , then the function (1.13) is strictly logarithmically completely monotonic with respect to ;

(3) for any given , the reciprocal of the function (1.13) is strictly logarithmically completely monotonic with respect to if and only if .our second result is presented in the following theorem.

For any given , let the function be defined on by



where denotes the Euler’s constant, then the function (1.14) is strictly logarithmically completely monotonic with respect to on . 

The following corollary can be derived from Theorems 2 immediately.

For any given , the inequality holds for all .

In order to prove our main results, we need the following lemmas.

It is well known that Bernoulli polynomials and Euler polynomials are defined by respectively [15].





The Bernoulli numbers are denoted by , while the Euler numbers are defined by .

In [16], the following summation formula is given:



for any nonnegative integer , which implies



In particular, it is known that for all we have



And the first few nonzero values are (see [17, p.804, Chapter23]).





The Bernoulli and Euler numbers and polynomials are generalized [18-21].

For real number and natural number , then

, , , only depend on natural number .

For real number and natural number , we have

For real number and natural number , we have

Let the sequence of functions for be defined on by 



the series is differentiable on , that is,

It is obvious that , therefore converges at . In order to prove (2.16), we need only to show that the inner closed uniform convergence of the series on . From (2.15), we have



For any interval , we have



for all . It is easy to check that the series converges, which and Weierstrass M-test implies that the series is inner closed uniformly convergent on . Hence the series is differentiable on and the identity (2.17) holds for .

The lemma is proved.

For and real number , let the function be defined by





If , then the function (2.19) satisfies

for all and .

Taking the logarithm of yields



and differentiating , then 



For given integer , we get



and, by the identities (2.13) and (2.14), (2.23) can be written as



Let and . It is easy to check that 



therefore is strictly increasing on , and then .

The following two cases will complete the proof of Lemma 5.

If , then since for , we have



which implies , and then for all .

If , then we get 



, , (2.27)

therefore is strictly increasing on , and then .

From (2.24), we know that the inequality (2.20) holds for and integer .

The lemma is proved. •

For and natural number , taking the logarithmically differential into consideration yields



where and stand for and respectively.

Furthermore, differentiating directly gives



Making use of (2.11) and (2.13) shows that for all and any fixed , the double inequality



holds for all and .

For any fixed , let and be defined on by respectively.

Differentiating and directly, we obtain





Therefore, for given we have



and



From (3.6) and (3.7), we conclude that for all we obtain



and



From (3.3) and (3.8)-(3.9), it is easy to see that



for all and all .

On the one hand, if , then the inequalities (3.10) can be equivalently changed into



and



for .

From (3.1), then simple computation shows that



for all and any given . As a result,



and 



for all and all .

Therefore, (3.14) and (3.15) imply



for all and all .

Hence, if either for given or for given , the function (1.13) is strictly logarithmically completely monotonic with respect to on , and if for given , so is the reciprocal of the function (1.13).

On the other hand, if for any given , then (3.10) implies



for .

In view of (3.13), we can conclude that



for . It is obvious that (3.18) is equivalent to that (3.14) and (3.15) hold for any given and . Therefore, it is easy to prove similarly that (3.16) is also valid on for any given and all .

The amino acid of antibody/protein possesses different preferences. Thus we can conduct site-directed mutation to promote the affinity and/or hydrophilic with the prediction of topology. For example, bovine antibodies have an unusual structure comprising a β-strand ‘stalk’ domain and a disulphide-bonded ‘knob’ domain in CDR3 (Figure 3). Attempts have been made to utilize such amino acid preference for antibody drug development.

 

 

Figure 3: Unique Structural Domain in Bovine IgG antibodies and application.

Consequently, the function (1.13) is the same logarithmically completely monotonicity on as on , that is, if either for given or for given , the function (1.13) is strictly logarithmically completely monotonic with respect to on , and if forgiven , so is the reciprocal of the function (1.13).

Conversely, we assume that the reciprocal of the function (1.13) is strictly logarithmically completely monotonic on for any given . Then we have for any given and all 



which implies



By L’Hˆospital’s rule, we have



for any given . By virtue of (3.20) and (3.21), we conclude that the necessary condition for the reciprocal of the function (1.13) to be strictly logarithmically completely monotonic is .

If the function (1.13) is logarithmically completely monotonic on 

for any given , then the inequality (3.19) and (3.20) are reversed for any given and all .

By utilizing (2.7) and (2.8), it is easy to see that



for any given . In fact, it is not difficult to show that the necessary condition for the function (1.13) to be strictly logarithmically completely monotonic is .

The proof of Theorem 1 is completed. •

Taking the logarithm of gives



Let 



then 



In view of Lemma 4, straightforward calculation gives



By virtue of (1.2), the identity (3.27) is equivalent to



By Lemma 5, we know that is strictly increasing on , which and (1.10) imply the limit of equals 1 as , therefore holds for all .

We know that is strictly completely monotonic on , where is defined by (1.8), hence for given integer , the inequality holds for all . 



And then by using inequality (1.9) and (1.10), we get for all .



From (3.29) and (3.31), we conclude that

for all . Utilizing Lemma 5 and (3.30), for given integer , it is easy to see that



for all .

Theorem 2 follows from (3.32) and (3.33).

Thus the proof of Theorem 2 is completed.