INTRODUCTION
Computational modeling (e.g., nuclear and engineering, business...etc) has
been rapidly growth within scientific researches and industry. During the process
of developments, new circumstances may be come to light (e.g., new problems,
areas of application and techniques appearing in modern dynamical processes
in physics, chemistry, engineering and economics), (SIAM
Working Group on CSE Education, 2001). Developers have produced numerous
novel mathematical algorithms aimed at stimulating many complex natural systems.
Therefore, to enhance the performance analysis of such computer simulations
an innovative ideas and foundational analysis for algorithms are required. Many
examples of such algorithms included in mathematical models and computer codes
in science and engineering can be found in literature including research of
business management (Taiwo, 2007; Mosa
et al., 2003) can work as a good example industry. In the meantime
(Yedjour et al., 2011) is a biocomputational
as well as neural network example. Furthermore, the new numerical algorithms
of (Ponalagusamy, 2008) can be considered as a good
example in the field of software engineering. The current study can be considered
as good example of computing mathematical methods in biology science based on
the use of green's functions (Bayin, 2006).
Cell membranes are multifarious systems, sensitive to various elements such
as environmental stress or the presence of other molecules besides containing
a rich mixture of lipids. They are able to adjust their shape and fluidity next
to another properties as well as a reaction to the new surroundings. Many questions
about how the properties of membranes are regulated still have to be answered
(Pandhal et al., 2008). The main objective of this
work was to investigate the effects of interaction between the proteins and
the lipid on the conformation and adsorption of the protein. The protein will
be modeled as copolymer of specific monomers. The choice of the amino acids
sequence distribution of the amino acids along the copolymer chain is to simulate
structure of the protein. This amino acids distribution will be used in the
numerical modeling. This will be established through:
• 
Computing the density profile ρ(z) of the protein by
using the Green's function technique (Bayin, 2006) 
• 
Computing the mean square endtoend distance of protein near the cell
membrane interface 
The theoretical and computational methods of this work are based on the basic
theoretical model developed by Khattari (1999). Diblock
copolymer system is very important and has been synthesized in several studies
(Baimark et al., 2008; Baimark
and Phromsopha, 2009). The adsorption behavior of diblock copolymer at interfaces
will be extended to study a more complex copolymers or proteins. In particular,
we will use the density profiles evaluated in the above reference and the mean
square endtoend distance of the copolymer, for further calculation which describes
the adsorption of the protein near the cell membrane. The conclusion and results
of Khattari (1999) were important for understanding the
properties of the biological macromolecules like protein at penetrable interfaces.
Where, such diblock copolymer exhibits stretching or elongation state which
is associated with the nonGaussian behavior so many systems can be derived
from the diblock copolymer system by studying the Gaussian behavior and controlling
some parameters like) χN (N is the number amino acids of the protein),
is the interaction parameter between the polymer and the cell interface), in
the limit (χN→∞) (strong segregation regime) the coil is stretched.
Such systems can be important in understanding the properties of the effect
of synthetic channel forming peptides on the phospholipid bilayer membrane structure
at penetrable interfaces.
ANALYTICAL MODIFICATIONS AND ITS CORRESPONDING VERIFICATION
The peptidemembrane interaction, channel forming and ion penetration through
the membrane channel can be studied by using the profile density approach p(z).
The equations (1, 2, 3, 4 and 5) have been reported by Khattari
(1999). The following set of equations present those equations, respectively:
where (p(z) and (p+(z) are obtained as:
The current updated analytical modifications and its corresponding verification can be started from Eq. 4. It reads the density of the diblock copolymer. Where, the composition, f = NA/ (NA+NB) and the radius of gyration, R^{2}_{g} = Nl^{2}/6.
COMPUTATIONAL AND MATHEMATICAL TECHNIQUES
Deriving the peptidelipid system equations: In this study, the protein will be modeled as copolymer of specific monomers for investigating the effect of protein and lipid interaction on the conformation and adsorption of the protein.
We adopted the amino acids sequence distribution of the amino acids along the
peptides chain to simulate the structure of the protein. Denote monomers as
(two components A and B) (polypeptide). On the other hand, denote copolymer
chain as (multicomponent peptide of portion chain) to simulate the idea of the
previous study (Khattari, 1999) with its extension (this
study).
Assumptions: To simulate the system let copolymer represents the protein, (copolymer = protein), Monomers (A and B) = Peptides A and B, (polypeptides) where (A and B ) are consists of two N = s sequences. Studying the system is depended on (N). N represents the number of amino acids sequence in (A and B) along the portion chain. NA = n1+n2 (2 sequence) NB = n3+n4 (2 sequence), In the meantime N total = n1+n2+n3+n4. (Polypeptide) = multicomponent peptide.
Consider a (multicomponent peptide chain of the protein consists of two components (A and B) each of them has two amino acids sequence of peptide (N = N1+N2+N3+N4) with several junction points (z_{0}, z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, …...z'), as illustrated in Fig. 1. (N) is the number of amino acids sequence in the peptides A and B of the protein chain.
Khattari (1999) addressed that the statistical weight
(G) is proportional to the number of conformations of the diblock copolymer
with ends fixed at z and z0. We can implements the same idea in our system,
where, the density profile of the system is equal to the number of conformations
of the ( multicomponent peptide) chain as a function of the position of the
joint points of theses conformations (peptides).

Fig. 1: 
Peptides (A, B) of the protein chain and its junction points 
Suppose that we have two conformations A and B, then the number of conformations
(density profile = P (Z)) is given by Eq. 6 adopted from the
Greens functions:
Where:
The normalization constant (Ω) is given by Eq. 8:
where, N_{A} = N1+N2, N_{B} = N3+N4 and N = N1+N2+N3+N4 and
substituting this into Eq. 6 will lead to Eq.
9:
Herein, we are searching to get an expression of p (z_{2}) like the
one appears in equation (Eq. 10), to match the following
(Khattari, 1999) equations:
Integrating the above equations with respect to (z', z_{1}, z_{0},
z_{2}, z_{3}) leads to the required expression of p(z_{2})
that stands for the density profile of the multicomponents peptide around the
junction (z_{2}) can be written by using Eq. 10:
On the other hand, for studying the stretching properties of the peptide chain
at the interface we calculate the mean square endtoend distance of the system
as a function of the distance to the interface by using the produced data of
the computed equation (Eq. 10). Simply the produced density
profile will be multiplied by (Z_{2})^{2} as illustrated in
equation (Eq. 11):
From now and then we will use P_{12} instead of P_{A} and P_{34} instead of P_{B}.
We need to derive an expression for both p_{12} and p_{34} starting from equation (Eq. 1) In order to compute the required density profile p(z) this leads to the following derivations:
• 
The density profile of p_{12} at z = z_{2 }where,
the total numbers of the amino acids in this (peptide) is N = N1+N2. Let
N_{1} = 1 and N_{2} = 1, Eq. 12 appears
in appendix 1 
• 
The density profile of p_{34} at z = z_{2 }where the total
numbers of amino acids on this peptide is N = N3+N4. Let N3 = 1 and N4 =1,
Eq. 13 appears in appendix 2 
Rewriting the above derived equations (Eq. 12 and 13)
into a simpler form in order to compute them enables us to get a new equation
sets (Eq. 14 and 15) besides which represents
the total density profile of the system p (z_{2}). Theoretical and computational
processes of the density profile are listed below:
The following new expressions of p_{12} and p_{34} can be written by substituting the intermediate values in to the derived equations:
Knowing that p_{12} and p_{34} have the same expressions by
following (Eq. 14 and 15) equations. As
well they may have the same value when N1 = N2 = N3 = N4 = 1. This means that
we can rewrite Eq. 11 by using p_{12} instead of
p_{34}, thus, P (total) = P (z_{2}) = (P_{12}) 2, But
changing the values of N = s will generate different values of p_{12,}
p_{34} and p (total), respectively. Consequently the values of p(z)
will be different for different compositions (F). On other words the statistical
weight (G) of this system is proportional to the number of conformations (amino
acids) of the peptide with ends fixed at z and z0, Fig. 1.
Program over viewing: This program is namely called (Ro calculation);
it provides a graphical user interface (GUI), works under the operating system
(windows). It produced tabulated data files. Appendix 3 defines
the class behaviors for the application. However, this program makes it easy
to specify the input file in order to schedule the computation processes, besides
viewing and saving the output data, therefore, it allows typical calculations
to be carried out far away from the deeply understanding of the theoretical
physics of this topic. The program has a source file written in the programming
language C++. Moreover, user has two options to run the program, each option
has a different set of input parameters that allows studying the density profile
under several preconditions related to the specified case study via the suitable
interface. Section three presents such case studies. Figures 2
and 3 present the two GUI interfaces of the program.
Selecting the first interface allows user to calculate the density profile
(p(Z_{2}), p_{12 },p_{34}) beside calculating the intermediate
values by inserting six parameters (Ω, Z_{2}, f, Rg, X, N),in this
case user has to set these parameters by following his specified case study
, almost it is a general case depended on (N) total (the total number of amino
acids in the multicomponents peptide chain (polypeptide) so that user can set
the parameter (f) to any value where 0< f <1.

Fig. 2: 
The main screen of the first option (general case) 

Fig. 3: 
The main screen plus the tabulated output file of the second
option 
Selecting this option means that user can get one output value for each density
profile (p(Z_{2}) , p_{12} ,p_{34}) then by using the
technique of (copy and paste) user can compose his own output data to be plotted
elsewhere . On the other hand Fig. 3 presents the main screen
of the second interface of the program which allows studying the effect of changing
the parameters (N1, N2, N3, N4) instead of studying the effect of N (total)
alone, where, N = N1+N2+N3+N4. Herein, the values of (f) will be changed following
the values of N1, N2, beside N3+N4 which affect the value of p12 and p34, respectively.
The input values panel is consists of two parts (begin values and end values)
in order to determine the interval of changing them. For example if user need
to calculate the values of (P_{12}, P_{34} , P (total)) when
10 ≤Z2≤ 10 he can set the begin value to 10 and the end value to 10,
after filling all other required fields and pressing the button (calculation)
the output file window will be opened automatically to show the tabulated data.
This file can be saved and imported to excel, origin or any related program
to be plotted. Next section includes more descriptions.
Program verifying and data validity: Here, we will generate data by
using the produced program all data and charts will be compared with those of
(Khattari, 1999).
Table 1: 
The data are generated by using the currently program (the
first interface of it) 

The produced data will be used for studying the interfacial properties of the
multicomponents peptide of the protein chain by calculating the density profile
(of the unit 106 mole cm^{3}) for several cases beside calculating
the mean square endtoend distance for studying the stretching properties of
the protein chain for several cases. This was established through:
• 
Calculating the density profile for different values of (f)
for the same value of (χ = 0.54) and different values of (χ) for
f = 0.5 
• 
Calculating the mean square endtoend distance for different values of
(f ) for the same value of (χ = 0.54) and different values of (χ)
for (f = 0.5) 
• 
Calculating the density profile for different values of (f) at the same
value of x 
Case one: Table 1: The data are generated by using
the currently program (the first interface of it), (Fig. 2).
Calculating the density profile as a function of the distance from the interface
z, where, z = 1, 0, 1 the density distribution of the system is shown for different
values of f (f = 0.3, 0.5, 0.7) for χ = 0.54, the case f = 0.5 is compared
with the MC results.

Fig. 4: 
The density profile as a function of the distance from the
interface z by using the currently program 
The values N = 32, Rg = 1, Ω = 1 are used in obtaining the density profiles.
Table 1 and Fig. 4 presents the results.
Data analysis: From Table 1 and Fig. 4, we can study the density distribution of the protein chain around z = 0, for different compositions f. (0.3, .0.5, 0.7). The density profile is symmetric for f = 0.50 with respect to the position of the interface z = 0, asymmetric and it is asymmetric for compositions f = 0.30 or 0.70.
Data validity: Figure 5 presents the density profile
as a function of the distance from the interface z, for different compositions
f (0.3, .0.5, 0.7) it has been adopted from (Khattari, 1999).
Generally our results show a very good agreement with both Khattari
(1999) and the prediction of Monte Carlo Simulations (MCS) Matsen
and Bates (1997) results. We verified this fact by comparing the density
profiles results of Fig. 4 with density profiles results of
Fig. 5. It is as expected, the density profile is symmetric
for f = 0.50 with respect to the position of the interface z = 0 and it is asymmetric
for compositions f = 0.30 or 0.70 in the Fig. 4 and 5.
in the most transparent symmetric case a comparison with the prediction of MCS
shows a very good agreement with our generated data as appeared in Table
1 when f = 0.5, (χ) = 0.54 around z = 0. Because the Monte Carlo Simulations
(MCS) predicts that the density profile in this status at the same conditions
where, f = 0.5 must be symmetric for composition f = 0.5. Figure
4 and 5 illustrate the same behavior of the relation between
p (z) and z/Rg, so that Leads to the same curve when plotting (p (z) versus
z/Rg) but not the same values of p (z) because (Khattari,
1999) has dealt with the density profile as a function of the distance from
the interface z, in units of wSSL where the radius of gyration in these units
is Rg = 4.2 wSSL.

Fig. 5: 
The density profile as a function of the distance from the
interface z, for different values of f(0.3, 0.5, 0.7) where χ = 0.54
and N = 32, as adopted from Khattari (1999) 
Whereas, Rg = 1 in all of our calculation. (Khattari, 1999)
also presented a comparison for the density profile with the Monte Carlo Simulations
(MCS) for the case f = 0.5 this comparison was in a good agreement with our
results.
Case two: Calculating the density profile as a function of the distance
from the interface z, where, z = (10, 0, 10). The density distribution of the
(p total, p_{12,} p_{34)} is shown for different values of f
(f = 0.001, 0.5, 0.999), respectively. Thus, for χ = 0.54, N = 32, Rg =
1, Ω = 1. Table 2 and Fig. 6, show
the results.
Table 2 show the data of case 2, generated by using the (first
interface of the program) (Fig. 1).
Data analysis: Figure 6 shows the density profiles of p_{12} and p_{34} for the strongly asymmetric composition (f = 0:999 or 0:001) and the density profile of P (total) with symmetric composition (f = 0:50).
Data validity: The Fig. 7 show that the asymmetry
of the composition plays an important role in determining the character of the
behavior of the peptide chain at interfaces. In the meantime that the protein
chain prefers the region where, the concentration of one kind of the amino acid
is rich. As a consequence of this rearrangement of the amino acids, the peak
of the distribution no more coincides with the interface position at z = 0.
For example, in the regime of strongly asymmetric composition the longer part
of the protein chain affects the density profile dramatically. In this case,
the whole profile appears to be very similar to a profile of a pure homopolymer
chain. Our results present a very good agreement with those of (Khattari,
1999) as appeared in Fig. 6.
Table 2: 
The data of case 2, generated by using the (first interface
of the program), (Fig. 1) 


Fig. 6: 
The density profile of (P (total), p_{12}, p_{34})
as a function of the distance from the interface z 

Fig. 7: 
The density profile of the whole system plus the subsystems
density profiles as a function of the distance from the interface z, in
units of wSSL, adopted from Khattari (1999), χ
= 0.54, f = 0.5, 0.001, 0.999., N=32, Ω = 1 
Another comparison of the symmetric case between our numerical evaluations
and those of (Khattari, 1999) is illustrated in Fig.
7.
Case 3: This case depends on generating data with respect to several
conditions in order to study the effect of changing the parameter (f). The generated
data are illustrated in Fig. 811 besides
Table 3 and 4.
By comparing our figures with those of (Khattari, 1999),
we found that the density profiles of the MCS are slightly broader than the
other density profile, nevertheless the agreement is very good.
Table 3: 
Data of Fig. 10 generated by using the
second part of the program (Fig. 3) 

Table 4: 
P (total), at different values of, f = 0.5, N = 32, Rg =
1, Ω = 1 


Fig. 8: 
The density profile of the whole protein chain as well as
the individual peptides (p_{12}, p_{34}) in the symmetric
case for =.54, Rg=1, Ω = 1, N=32 

Fig. 9: 
The density profile as a function of the distance from the
interface z, in units of wSSL, the density distribution of the whole system
(PAB) are shown as well as the individual subsystems (PA and PB) and compared
with the predications of the MC results for the symmetric case for = 0.54.
Adopted from Khattari (1999) 
The density profiles of the whole system (p (total) = p_{12}*p_{34})
and of the individual profiles (p_{12}, p_{34}) for different
compositions (f = 0:50, 0:40 and 0:60) are shown in Fig. 8.

Fig. 10: 
The density distribution of the whole system and the individual
subsystems for f=0:5, 0:4 and 0:6 for = 0:54, Rg=1, N=32, Ω = 1(Table
3) 

Fig. 11: 
The density profile as a function of the distance from the
interface z, in units of wSSL, the density distribution of the whole system
and the individual subsystems for f = 0:5, 0.4 and 0.6 for x = 0.54. Adopted
from Khattari (1999) 
The Fig. 11 show that the asymmetry of the composition plays
an important role in determining the character of the behavior of the peptide
chain at interfaces.
In Table 4 p(total), at different values of, f = 0.5, N =
32, Rg = 1, Ω = 1
Figure 12 p(total) versus z/Rg, where, Rg =1, Ω =1, f = 0.5
Calculating the Density Profile for Different Values of (χ) at the
Same Value of (f): The dependence of the interfacial profile on the interaction
parameter χ is illustrated in Fig. 1213
and Table 4. The profiles are shown for χ = 0.54, 0.75
and 0.85 at symmetric composition f = 0.50. It shows how the distribution profile
of the protein chain is affected by the variation of the interaction between
the peptides and interface.

Fig. 12: 
P (total) versus z/Rg, where, Rg =1, Ω =1, f = 0.5 

Fig. 13: 
The density profile as a function of the distance from the
interface z, in units of wSSL, the density distribution of different values
of x, for f = 0.5, adopted from Khattari (1999) 
From the Fig. 12 and 13 we can noted
that increasing the value of χ makes distribution narrower and vice versa.
This can be interpreted due to the effect of localization of the peptide_protien
at the interface. A common feature of all the above profiles is that the peaks
of the distributions are broader than the gyration radius Rg. The above results
agree with the experiments of Dai et al. (1994)
and Russell et al. (1991) and with the simulations
of Werner et al. (1996) plus the mean field theory
of Noolandi and Hong (1982).
The stretching properties of the peptide chain: In this section, we
consider the stretching properties of the protein chain (peptides+amino acids).
Figure 14 and 15, illustrate the chain
stretching in the direction perpendicular to the interface. This Fig.
14 and 15 shows the zcomponent of the mean square endtoend
distance obtained by using Eq. 11 after computing the density
profile for different compositions (f = 0.50, 0.45 and 0.55).

Fig. 14: 
The mean square endtoend distance by using the current program,
for χ = 0.54, different values of f (0.54, 0.5, 0.55), Rg = 1, N =
32.Ω = 1 

Fig. 15: 
The mean square endtoend distance obtained by using the
current program (second interface), for χ = 0.54, different values
of f (0.54, 0.5, 0.55). Rg = 1, N = 32 but N1 = 8, N2 = 8, N3 = 8, N4 =
8, Ω=1 
The figures are shown a good agreement with Fig. 16 that
was adopted from Khattari (1999). Moreover, this picture
was confirmed in many previous investigations such the experiments of Sommer
and Daoud (1995) and by simulations Sommer et al.
(1996), in addition to the study of Werner et al.
(1996).

Fig. 16: 
The mean square endtoend distance in unites of a2N = 3,
plotted vs. z=wSSL. The results are shown for different values of f for
χ = 0.54, the symmetric case is compared with the MC results. Adopted
from, Khattari (1999) 
The effect of the stretching is very strong for those peptides that centered
at about two to three gyration radii away form the interface but it almost disappears
for the peptides that centered at the interface.
The mean square endtoend distance by using the current program, for = 0.54, different values of f (0.54, 0.5, 0.55). Rg = 1, N = 32, Ω = 1 (Fig. 14).
The mean square endtoend distance obtained by using the current program (second interface), for = 0.54, different values of f (0.54, 0.5, 0.55). Rg =1, N = 32 but N1= 8, N2 = 8 ,N3 = 8 ,N4 = 8, Ω =1 (Fig. 15).
The mean square endtoend distance in unites of a2N = 3, plotted vs. z = wSSL
(Fig. 16). The results are shown for different values of
f for = 0:54, the symmetric case is compared with the MC results. Adopted from,
Khattari (1999).
The effect of the interaction of the peptide components with the interface
is illustrated in Fig. 17, the data of mean square endtoend
distance was generated by using Eq. 11 after computing the
density profile of this case for three different values of the interaction parameter
χ = 0:54, 0:65 and 0:75 for two compositions f = 0:50 and 0:40. Our results
show a good agreement with Khattari (1999) results that
is illustrated in Fig. 18. It is understandable from Fig.
18, that the central region and the wings of the protein chain density profiles
(p_{12}, p_{34}) are hardly affected by increasing the value
of the interaction parameter between the peptide parts and the interface. On
the other hand the amount of the peptides located at about two to three gyration
radii is more affected by this variation of the interaction parameter. Increasing
the value of (χ) results in stretching the protein chain coil in its rich
phase.

Fig. 17: 
The mean square endtoend distance the results are shown
for different values of χ for f = 0.5 and 0.4. by using the currently
program 

Fig. 18: 
The mean square endtoend distance in unites of a2N = 3,
plotted vs. z = wSSL. The results are shown for different values of x for
f = 0.5 and 0.4, adopted from Khattari (1999) 
From these results we can say that peptide chain behaves as it would be composed
of two independent parts that associated with the nonGaussian behavior when
stretching. However, this is associated with the chain stretching. The existence
of the chain stretching is clear hint that the system exhibits a crossover from
the weak segregation to the strong segregation regimes. In the weak segregation
regime, where it is assumed that the chain is Gaussian coil, the structure is
not affected by the parameter χN (N is the number amino acids of the protein,
χ is the interaction parameter between the polymer and the cell interface)
in contrast, in the limit (χN→∞) (strong segregation regime) the coil
is stretched. Such systems can be important in understanding the properties
of the biological macromolecules like protein at penetrable interfaces (Muller
and Schmid, 1998).
DISCUSSION
The density profile ρ(z) and mean square endtoend distance R(z) of the
protein near the cell membrane interface have been effectively computed by using
the programming language C++ (MS VS) based on the Green's function technique
adopted from Khattari (1999). The modified model has
been successfully used in generating data easily. The analytical modifications
and its corresponding verification has been discussed, results and charts were
illustrated and compared with those of literate. The program allowed a rapid
processing and viewing of the calculated data for all density profiles (p (z_{2}),
p_{12}, p_{34}), in addition to all intermediate parameters
that might be used in the calculations. In order to determine the accuracy of
the calculations and enable testing the results with respect to several preconditions,
the program provided an access to many parameters such as (N, Rg, Ω, (χ),
Z, f, N_{1}, N_{2}, N_{3}) . Input parameters can be
easily entered through a friendly graphical user interface (GUI), however, the
model has two different interfaces and each one has its own function for saving
user time and verifying the output data. Outputs data has two types a single
value of each density profile (p_{12},p_{34},p(z_{2}))
and a set of values for each density profile following the specified rang of
the input parameters (e.g., 10≤0≤10 ) .The scope of the future work is
to add some other models(e.g., protein chain may consist of (A, B, C, D…)
component (peptides). In general, the generated data showed a very good agreement
with both Khattari (1999) and the prediction of Monte
Carlo Simulations (MCS) Matsen and Bates (1997) results.
The current study addressed the following results:
• 
Changing the values of N’s (amino acids) will generate
different values of p_{12 }, p_{34} as well as p (total),
respectively. Consequently the values of p (z) will be different for different
compositions (f). In other words the statistical weight (G) of this system
is proportional to the number of conformations of the protein chain with
ends fixed at z and z_{0} 
• 
The asymmetry of the composition played an important role in determining
the character of the behavior of the peptide chain at interfaces. Also the
protein chain (peptide+amino acid) preferred the region where the concentration
of one kind of the amino acids is rich. As a consequence of this rearrangement
of the amino acids, the peak of the distribution no more coincides with
the interface position at z = 0 
• 
As a result of studying the effect of changing χ increasing the value
of χ makes distribution narrower and vice versa. This can be interpreted
due to the effect of localization of the peptide at the interface. A common
feature of all tested profiles was that the peaks of the distributions are
broader than the gyration radius Rg 
Finally, the program Ro calculation should be useful in rapidly providing large
amount of data for modeling peptide_ lipid system.
CONCLUSION
The current study has been successfully investigated the effects of interaction between the proteins and the lipid on the conformation and adsorption of the protein by using a MS Visual Studio C++ computer program that has been computed based on using the same mathematical techniques of a literature model of interfacial properties of diblock copolymers at penetrable interfaces but with diffident assumptions. To study system more complex proteins the new model can be expanded to include larger systems of more than two amino acids for (N = N_{A}+N_{B}+N_{C}+N_{D}YYNN); where, N > N_{1}+N_{2}+N_{3}+N_{4}.
FUTURE WORK
• 
Add plotting capabilities to the excitants numerical model 
• 
Extend the model to include larger systems that may include more than
two amino acids for N = N_{A}+N_{B}+N_{C}+N_{D}……NN
where, N > N_{1}+N_{2}+N_{3}+N_{4} 
APPENDIX
Appendix 1: Equation (Eq. 12)
Appendix 2: equation (Eq. 13)
Appendix 3 


