Analysis Of Antibody By Real-Valued Special Functions

Background: Along with the rapid development of genetic engineering technology and antibody engineering technology, humanized monoclonal antibody has been rapidly developed and gradually replaces the rat sourced monoclonal antibody. In this paper, we establish two new logarithmically completely monotonic functions involving the real-valued special functions according to two preferred interaction geometries, necessary and sufficient conditions are presented for one of them to be logarithmically completely monotonic. As a consequence, a sharp inequality involving the real-valued special functions is deduced to solve the problems of genetically engineered antibody. Abstract

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.  We next recall [2][3][4][5] that a function f is said to be completely monotonic on an interval I , if f has derivatives of all orders on I which alternate successively in sign, that is, where ( ) t µ is non-decreasing and the integral converges for 0 x > .
We recall also [7][8][9] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and for all x I ∈ and for all 1 n ≥ . If inequality (1.7) is strict for all x I ∈ and all 1 n ≥ , then f 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 ( ) 0, ∞ , with two limits (1.9) From (1.9) and the monotonicity of ( ) g x , then the double inequalities ( ) holds for all 0 x > .
In [11,Theorem 1], by using the well-known Binet's formula, H. Alzer generalized the monotonicity and convexity of ( ) g x , that is, the function is strictly completely monotonic on ( ) 0, ∞ if and only if In [12], D. Kershaw and A. Laforgia proved that the function ( ) These are equivalent to the function ( After the papain digestion, the remained antibody functional part (usually the Fab domain), will be smaller and the structure is also changed (Figure 1b) 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. (1.13)

Theorem 1
For the function (1.13), then the following statements are true: 1. for any given where γ denotes the Euler's constant, then the function (1.14) is strictly logarithmically completely monotonic with respect to x on The following corollary can be derived from Theorems 2 immediately.

Corollary 1
For any given 1 y ≥ , the inequality holds for all 0

Lemma
In order to prove our main results, we need the following lemmas. It is well known that Bernoulli polynomials ( ) k B x and Euler polynomials ( ) k E x are defined by respectively [15].
The Bernoulli numbers n B are denoted by In [16], the following summation formula is given: for any nonnegative integer k, which implies In particular, it is known that for all n ∈  we have And the first few nonzero values are (see [17,p.804,Chapter23]).

Lemma 2 [24, Lemma 3]
For real number 0 x > and natural number n , we have Lemma 3 [1,17] For real number 0 x > and natural number n, we have

Lemma 4
Let the sequence of functions

Lemma 5
For 0 1 a < ≤ and real number b, let the function ( )

Proof
Taking the logarithm of ( ) Making use of (2.11) and (2.13) shows that for all n ∈  and any fixed 0 y > , the double inequality for n ∈  . It is obvious that (3.18) is equivalent to that (3.14) and (3.15) hold for any given and ( ) , 0 x y ∈ − . Therefore, it is easy to prove similarly that (3.16) is also valid on ( ) , 0 x y ∈ − for any given 0 y > and all n ∈  .
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. Conversely, we assume that the reciprocal of the function (1. 13 The proof of Theorem 1 is completed.

Proof of Theorem 2
Taking the logarithm of ( ) x n x y n x y

Conclusion
In conclusion, we establish two new logarithmically completely monotonic functions involving the gamma function according to two preferred interaction geometries, and a sharp inequality involving the gamma function is deduced to solve the problems of genetically engineering antibodies. It is necessary to address, many other aspects (such as thermal condition, alkalinity or acidity, adhesion of antibodies) are also playing key roles in antibodies functioning, which could be also understood from bio-mathematical perspective, and such knowledge will be in return useful for biomedical application of antibodies as well as proteins [25][26][27][28][29][30].