Mathematical Analysis of Antibody by Gamma and PSI Functions

Mathematical or computational models have been proposed in different biological subjects in order to explain various complex phenomena. We have designed this study to monoclonal antibodies and Fc-fusion proteins (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 complementarities 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. Abstract


a.
Diagram of infliximab (whole IgG1 molecule against TNFα ). Red: VH and VL domains; Blue: constant domains of heavy and light chains. Therapeutic IgGs may differ regarding the heavy-chain iso type and the degree of variable domain humanization. For instance, Ixekizumab is a humanized IgG4 and denosumab a fully human IgG2. b.
Etanercept is a fusion protein that combines the TNF receptor to a human IgG1 Fc fragment. Dark red: cysteine-rich domains (CRD) of the TNFα receptor. c.
Certolizumab pegol is a pegylated Fab fragment. Green: polyethylene glycol chains. Glycosylated proteins (N-glycans and O-glycans) are not shown, to improve readability. Where, 0.577215664901 γ =  denotes the Euler's constant. 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 target (Figure 2), i.e.: antigen, receptor. How to describe the changes in the view of topology? The following cases will explain it in detail.
Diagrammatic representation of VH germ-line diversity (top) leading to domain structure diversity (bottom) (Figure 2). In humans, multiple V, D, J regions are rearranged to produce a VDJ segment that encodes the variable region (in this case the VH). The assembled polypeptide is represented by the classically 'Y' shaped IgG (150kDa) molecule (bottom), where the various regions of the antibody are labeled and the carbohydrates are indicated by pink dots. Unlike humans or mice, chickens use gene conversion to diversify the antibody repertoire. Single genes encode the VH (and VL) and a single VDJ event, which generates minimal diversity, is preceded by insertion of upstream pseudogenes (VH) in a process that results in donation of diverse sequence fragments into the V region. Chicken antibody (IgY) has a molecular weight of 180 kDa by virtue of an additional constant heavy domain and additional carbohydrate moieties. Cows employ an ultra long CDRH3 repertoire resulting from a VDJ event that utilises a single VH segment (VH Bul), which combines with an elongated D region (DH2) to produce a recombined V-region that undergoes diversification by somatic hyper-mutation. This can introduce or remove cysteine thus, diversifying the disulfide landscape of the repertoire.

Approaches in Poultry, Dairy & Veterinary Sciences
It was proved explicitly in [8] and other articles that a logarithmically completely monotonic function must be completely monotonic. In [10] From (1.9) and the monotonicity of ( ) g x , then the double inequalities ( ) 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 ( In [12], D. Kershaw and A. Laforgia proved that the function ( ) These are equivalent to the function ( ) being decreasing on ( ) 0, ∞ , respectively. In Theorem 5, F Qi & Ch-p Chen [13] generalized these functions. They obtained the fact that for all is strictly increasing for 0 r ≥ and strictly decreasing for 1 r ≤ − , respectively. After the papain digestion, the remained antibody functional part (usually the Fab domain), will be smaller and the structure is also changed. These changes can be revealed vividly using topology. Recently [14,15], Theorem 1, F Qi, et al. [14] established another excellent result, which states that forgiven Antibodies occur spontaneously gathering and forming dimer, polymer, which will influence their functions ( Figure 3). 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 ( ) 0, y ∈ ∞ and real number α , let the (1.13)

Our First Result is Contained in the Following Theorem
iii. For any given 0 y > , the reciprocal of the function (1.13) is strictly logarithmically completely monotonic with respect to Our second result is presented in the following theorem.

Theorem 2: 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 ( ) 0, ∞ . The following corollary can be derived from Theorems 2 immediately.

Lemma
In order to prove our main results, we need the following lemmas. It is well known that Bernoulli polynomials , while the Euler numbers n E are defined 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 .

Lemma 2: [24]
For real number 0 x > and natural number n , . It is easy to check that the series The lemma is proved.

Lemma 5:
For 0 1 a < ≤ and real number b , let the function  The following two cases will complete the proof of Lemma 5. The lemma is proved.

Proof of Theorem 1
For 0 x ≠ and natural number n , taking the logarithmically differential into consideration yields Making use of (2.11) and (2.13) shows that for all n ∈  and any fixed 0 y > , the double inequality For any fixed for all k ∈  and all 0 x > . Therefore, (3.14) and (3.15) imply for n ∈  . In view of (3.13), we can conclude that for n ∈  . It is obvious that (3.18) is equivalent to that (3.14) and (3.15) hold for any given

Appro Poult Dairy & Vet Sci
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. Unique Structural Domain in Bovine IgG antibodies and application ( Figure 3).

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 antibody. It is necessary to address, many other aspects (such as thermal condition, alkalinity or acidity, adhesion of antibody) are also playing key roles in antibody 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.