A proper cognition about the osmotic force per unit area and thermodynamic behavior of protein solutions is critical for planing an efficient protein separation procedure. It is besides of great importance to develop a rapid and cheap technique to accurately gauge the protein osmotic force per unit area. An intelligent theoretical account based on the feed-forward unreal nervous web ( ANN ) to gauge the osmotic force per unit area of bovine serum albumen ( BSA ) in footings of pH, ionic strength and BSA concentration is proposed in this paper. Osmotic force per unit area of BSA is besides modeled through the application of a colloidal interaction attack. Molecular interaction forces such as electrostatic, London-van der Waals, and hydration along with entropy force per unit area are considered in the colloidal theoretical account to foretell the BSA osmotic force per unit area. The ANN anticipations were compared with the consequences obtained from the colloidal interaction theoretical account and experimental informations. Good understanding was observed between the predicted osmotic force per unit area values and the experimental information. It is besides concluded that the ANN technique exhibits higher truth in foretelling the osmotic force per unit area of BSA for broad scopes of input variables if 8 nerve cells are selected for the concealed bed in the ANN construction. Consequences of this survey indicate that ionic strength and pH among the input parametric quantities selected for the ANN have the greatest impacts on the osmotic force per unit area value. The proposed ANN theoretical account serves as a dependable tool for fast, low cost and effectual appraisal of osmotic force per unit area in the absence of equal experimental informations.

Keywords: Osmotic Pressure, Artificial Neural Network, Bovine Serum Albumin, Molecular Interaction, Back Propagation

## 1. Introduction

Osmotic force per unit area of protein molecules plays an of import function in design and scale-up of separation and purification procedures such as chromatography and membrane filtration. In a membrane separation procedure, the concentration of the maintained solutes at the membrane surface may make impregnation degrees which can take to elevated osmotic force per unit areas. The high osmotic force per unit area at the membrane surface decreases the efficiency of the procedure by take downing the permeate flux. Therefore, accurate anticipation of osmotic force per unit area of protein solutions particularly at high concentrations is important in gauging permeate flux during membrane filtration processes ( Howell et al. , 1993 ) .

The magnitude of osmotic force per unit area depends on the physicochemical belongingss of the solution such as pH, ionic strength, protein type and concentration. A thermodynamic relation for osmotic force per unit area can be derived from the Gibbs free energy equation and the construct of chemical potency ( Hiemenz and Rajagopalan, 1977 ) . Sing the premises of ideal and incompressible solution, pertinence of Virginia n’t Hoff ‘s equation obtained from the Gibbs free energy equation is limited to thin solutions. Based on the Virginia n’t Hoff ‘s equation, osmotic force per unit area increases linearly with solute concentration. For concentrated solutions, the Virginia n’t Hoff ‘s equation can be modified utilizing virial equations as the coefficients of the equation are obtained by suiting the abbreviated virial equation to the experimental information. Harmonizing to statistical mechanics, the 2nd virial coefficient corresponds to the interactions between atom braces, while higher orders of virial coefficients are associated with larger figure of atoms ( Cheryan, 1998 ; Everett, 1988 ) . The major restriction of this method is that the experimental informations is normally scarce in the literature and confined to a dilute scope.

Assorted experimental techniques exist to find the osmotic force per unit area. Osmotic force per unit area is by and large measured by a membrane osmometer that is made of two Chamberss, separated by a semi-permeable membrane ( permeable merely to the dissolver ) ( Moon et al. , 2000 ) . Another common method that estimates the osmotic virial coefficient is the inactive visible radiation dispersing ( SLS ) technique ( Ahamed et al. , 2005 ) . The SLS is normally used to obtain the molecular weight of the solute and the 2nd virial coefficient. Due to its expedience and physiological importance, bovine serum albumen ( BSA ) in aqueous solutions has been widely employed in surveies related to membrane ultrafiltration procedures. However, really few comprehensive probes have been conducted on the BSA osmotic force per unit area at high concentrations. The most of import research in this respect has been done by Vilker et Al. ( 1984 ) . Osmotic force per unit area of BSA up to a protein concentration of 475 g/L was measured in their survey by a membrane osmometer for limited scopes of pH and ionic strength. A considerable divergence from ideality was experienced even at moderate concentrations of protein. A strong relationship between the osmotic force per unit area and solution pH was observed. Kanal et Al. ( 1994 ) examined the consequence of pH on the osmotic force per unit area for a dilute scope of BSA concentration up to 100 g/L in 0.1M NaCl buffer solution. Osmotic force per unit area was found to be minimal at pH of 4.6 and increased by a factor of five when pH dropped from 4.6 to 3. The fluctuation of osmotic force per unit area with pH was explained by the alteration in the protein conformation and collection.

The construct of molecular interactions was employed by assorted research workers ( Wu and Prausnitz, 1999 ; Lin et al. , 2001 ; Bowen et.al, 1995a, 1996a, ) to foretell the osmotic force per unit area of BSA. Wu and Prausnitz ( 1999 ) developed two new waves der Waals type theoretical accounts and compared the consequences against experimental osmotic force per unit area of dilute BSA solutions ( & lt ; 150g/L ) within the scopes of 4.5-7.4 and 1-5 M for pH and Na chloride concentrations, severally. In both theoretical accounts, the potency of average force between proteins consisted of hard-sphere repulsive force, new wave der Waals attractive force, and double-layer repulsive force. The part of the difficult sphere repulsive force to the osmotic force per unit area was represented by the Carnahan-Starling equation of province ( EOS ) in both theoretical accounts. The 2nd virial coefficient in the first theoretical account and the random-phase-approximation ( RPA ) of the 2nd theoretical account represented the new wave der Waals and double-layer interactions. Neither theoretical account was successful in foretelling the BSA osmotic force per unit area at high salt concentrations. A more precise potency of average force was recommended for molecular mold of osmotic force per unit area of protein molecules in salt solution. Lin et Al. ( 2001 ) proposed an equation of province to foretell the BSA osmotic force per unit area with one adjustable parametric quantity. They expanded the Duh and Mier-Y-Teran equation of province for one Yukawa possible to two Yukawa potencies to stand for the abhorrent interaction and the attractive interaction between charged BSA molecules. The Carnahan-Starling equation was used to cipher the difficult sphere abhorrent interaction. The adjustable parametric quantity was the scattering energy parametric quantity in the Yukawa potencies alternatively of Hamaker invariable in the Derjaguin-Landau-Verwey-Overbeek ( DLVO ) theory and the parametric quantity of minimal distance between two protein surfaces. Their theoretical account was non valid when the BSA concentration was high or/and asymmetric microions existed in the solution. A colloidal interaction theoretical account was developed by Bowen et.al ( 1996a ; 1995a ) to cipher the osmotic force per unit area of colloidal system using the drawn-out DLVO theory. The DLVO theory considered a additive combination of London-van der Waals attractive and electrostatic repulsive forces as a map of distance separating atoms ( Hunter, 2001 ) . The multiparticle nature of the electrostatic interaction between protein molecules was taken into history utilizing the Wigner-Seitz theoretical account. The new wave der Waals attractive force force was estimated utilizing the Lifshitz-Hamaker invariable. Good understanding between the colloidal theoretical account consequences and the experimental BSA osmotic force per unit area was found for the studied operating scope.

Artificial nervous web ( ANN ) theoretical accounts are potentially dependable tools for the appraisal of complex parametric quantities such as osmotic force per unit area if the web is successfully trained. Many interconnectednesss in the ANN offer a immense grades of freedom or adjustment variables. Therefore, ANN enables to exemplify the non-linearity of a system in contrast to conventional techniques. Another benefit of ANNs is that they are dynamically adaptative where they can larn and set to new conditions in which the efficiency of ANN is non sufficient with old state of affairss. Furthermore, the ANN theoretical account is able to manage systems with several inputs and end products.

In this paper, unreal nervous web ( ANN ) and colloidal interaction theoretical account were employed to foretell osmotic force per unit area of BSA. Osmotic force per unit area was considered as a map of pH, ionic strength, and BSA concentration. The colloidal interaction theoretical account besides included the consequence of thermodynamic belongingss on the magnitude of BSA osmotic force per unit area. In the ANN theoretical account, the experimental information from the literature was divided into three classs, including preparation, proof, and proving datasets ( Maier and Dandy, 2000 ) . Levenberg-Marquardt optimisation algorithm was used for developing the web. The optimal ANN construction was determined through minimising the absolute per centum mistake ( ANN theoretical account anticipations and experimental informations ) and the absolute comparative difference between the values calculated by ANN and colloidal interaction theoretical accounts. The public presentation of the mathematical theoretical accounts presented in this survey was checked based on statistical parametric quantities, viz. R-squared, upper limit and intend absolute per centum mistakes. Reasonable understanding between the consequences was obtained which demonstrated the utility of the ANN theoretical account in anticipation of BSA osmotic force per unit area. The methodological analysiss used in this survey and the consequences obtained are discussed in inside informations throughout the undermentioned subdivisions.

## Artificial Neural Network

Artificial nervous web ( ANN ) is a robust technique which is able to capture and stand for complex relationships between inputs and end products. ANN has been inspired from the information processing mechanisms of the encephalon ( Zurada, 1992 ; Murray, 1995 ) . Harmonizing to William James ( 1980 ) “ the sum of activity at any given point in the encephalon cerebral mantle is the amount of the inclinations of all other points to dispatch into it, such inclinations being proportionate 1 ) to the figure of times the exhilaration of other points may hold accompanied that of the point in inquiry ; 2 ) to the strengths of such exhilarations ; and 3 ) to the absence of any rival point functionally disconnected with the first point ” . This thought enabled McCulloch and Pitts ( 1943 ) to explicate the first theoretical accounts of a biological nerve cell. Over the last few old ages, ANN has been loosely employed for legion applications such as procedure control, behavior anticipation, theoretical account acknowledgment, and system categorization.

An ANN theoretical account is composed of a big figure of interrelated nerve cells which can be considered as a calculating engine that receives inputs, processes them in a concealed bed and so generates an end product. Figure 1 depicts a simple architecture of an ANN with one hidden bed. The Numberss of nerve cells and concealed beds depend on the complexness and nonlinearity of the job. Each nerve cell is connected to the input and end product by a corresponding weight. Inputs to each nerve cell are multiplied by their corresponding weights ( IW ) . After that, the merchandises are summed up together and with a prejudice nerve cell ( b1 ) , and the amount is processed utilizing a nonlinear transportation map such as inflated tangent sigmoid ( tansig in MATLABA® ) to obtain a1 ( Torrecilla et al. , 2005 ) . The produced matrix, a1, is subjected to layer weights ( LW ) and the prejudice, b2. The merchandise is applied to a additive transportation map ( purelin in MATLABA® ) to make an end product. Initial values of weights are indiscriminately assigned by using unvarying or Gaussian distribution. There are a figure of different nervous web constructions and larning algorithms ( e.g. , back extension ( BP ) and multiple bed perceptron ( MLP ) ) for the ANN technique. The BP method is recognized as one of the most general acquisition algorithms. The BP technique is employed in the feed-forward ANN. It implies that the nerve cells are arranged in beds, and convey their signals “ frontward ” , and so the mistakes are propagated backwards. The BP algorithm includes a preparation procedure in which a series of inputs and end products is provided and the web predicts the end products based on the randomly assigned initial weights. Then the mistake ( average squared mistake ) between existent and predicted consequences is computed. The weights are modified through the preparation procedure until the mistake between the existent end product and the predicted end product is minimized. The BP algorithm adjusts the weights utilizing the gradient descent rule where the alteration in the weight is relative to the mistake gradient with a negative mark ( Alshihri et al. , 2009 ; Hagan and Beale, 1996 ) . The proof set prevents overfitting of the web by halting preparation one time MSE in the proof set begins to increase. The public presentation of the ANN theoretical account is eventually checked by presenting new and independent datasets to the trained theoretical account.

An option to the MLP is the radial footing map ( RBF ) web. The RBF web is recommended for map estimate jobs which have local lower limit. The RBF type ANN guarantees convergence to globally optimal parametric quantities. Furthermore, RBF networks execute more robustly, compared to MLP webs when noised input informations set are involved in ANN. For the jobs without local lower limit and for categorization jobs, MLP networks with BP developing algorithm are preferred.

## Colloidal Interaction Model

The methodological analysis proposed by Bowen et Al. ( 1996a ; 1995a ) was employed in this survey to gauge the osmotic force per unit area of BSA for broad scopes of pH, ionic strength, and BSA concentration. Table 1 summarizes the equations which were used in the colloidal interaction theoretical account. Electrostaticss ( FELEC ) , London-van der Waals ( FATT ) , and hydration forces ( FHYD ) every bit good as entropic force per unit area ( PENT ) were considered in the drawn-out DLVO theory to gauge the osmotic force per unit area ( Equations 1-4 ) . The modified Gouy-Chapman electrical dual bed theoretical account ( EDL ) was used to depict charge and possible distribution around the charged BSA molecules in the electrolyte ( Bowen and Williams, 1996a ; Hunter, 1993 ; Brett, 1993 ) . In the EDL theoretical account, electrical dual bed was formed by a compact bed of hydrous counterions around the protein surface followed by a diffuse bed widening into the bulk solution. Since the charge on the protein surface was non to the full compensated by the compact bed, extra ions were attracted to the surface with weaker electrostatic forces. The ion distribution was expressed by the Poisson Boltzmann equation ( Equations 5-7 ) . When charged atoms approach one another, their diffusion beds start to overlap, taking to a abhorrent force which prevents farther intimacy. The multi-particle nature of such interactions was considered utilizing a Wigner-Seitz cell theoretical account ( Figure 2 ) ; each cell was presumed to be comprised of a individual charged atom surrounded by a shell of fluid ( Bowen and Jenner, 1995a ; Wigner and Seitz, 1934 ) . The effectual country occupied by the protein at a conjectural plane ( surface of a hexangular cell ) , Ah, was calculated from Equations 2-4. The parametric quantity is the volume fraction of the protein and was calculated from Equation 4. Surface charge and zeta potency of the protein molecule were calculated based on the charge ordinance theoretical account. The charge ordinance theoretical account required the type and figure of the amino acid groups of the protein participated in the ionisation reaction and besides the sum of ions adsorbed on the protein surface ( Bowen and Williams, 1996a ) .

Equation 9 represents the London-van der Waals energy between two similar size spherical atoms with their centres a distance ( Dp+2a ) apart. Molecules in close propinquity induce charge polarisation due to the electromagnetic fluctuations. These forces grouped as London-van der Waals forces are inherently attractive. These attractive forces can go effectual when the surfaces approach one another. The new wave der Waals forces ( Equations 8-9 ) required the appraisal of the Hamaker invariable for BSA which was obtained from the refractile index informations of BSA in the solution and Lorenz-Lorentz equation ( Bowen and Williams, 1996b ; Hough and White, 1980 ; Bowen and Jenner, 1995b ; Nir, 1977 ) . Other abhorrent forces between proteins are the hydration forces frequently referred to as polar interactions ( Equation 10 ) . Strong polar interactions orient H2O molecules adsorbed on the surface of proteins, and therefore the stableness of the colloidal system is conferred by those hydrous H2O molecules that force the two proteins apart at contact. Entropic force per unit area ( Equation 11 ) was calculated utilizing an equation proposed by Hall, offering the best difficult domain entropic force per unit area consequences for both high and low volume fractions ( Hall, 1972 ) . Technical readers are encouraged to analyze the undermentioned mentions for more information ( Bowen and Jenner, 1995a ; Bowen and Jenner, 1995b ; Wigner and Seitz, 1934 ; Bowen and Williams, 1996a ; Bowen and Williams, 1996b ; Hough and White, 1980 ) .

## Consequences and Discussion

## ANN Model

The ANN theoretical account included three independent variables ( pH, ionic strength and protein concentration ) . As was discussed before, the magnitude of the osmotic force per unit area is controlled by these three parametric quantities ( Bowen and Williams, 1996a ; Vilker et al. , 1984 ) . Input information was normalized before get downing the ANN mold to avoid any false influence of factors with higher order of magnitude. Data standardization was performed utilizing Equation ( 12 ) as follows:

where xmax and xmin are the highest and lowest values of variable ten, severally. The ANN theoretical account was trained utilizing experimental informations from literature ( Vilker et al. , 1984 ; Wu and Prausnitz, 1999 ) . The preparation and proof datasets were selected indiscriminately from the available osmotic force per unit area data.A MLP web was used in the current survey to gauge osmotic force per unit area. The conventional back extension method was used to modify the weights. Osmotic pressureA in footings of input parametric quantities selected in this survey does non hold multi-peak nonlinear maps ( Vilker et al. , 1984 ; Wu and Prausnitz, 1999 ) . Therefore, there is no concern of running into local optima instead than planetary optima ( It is deserving adverting once more that RBF webs are recommended for jobs with local lower limit ) . Therefore, conventional back extension is an effectual preparation algorithm that can be used with no hazards of disconvergence. Execution of different ANN theoretical accounts and comparing of their public presentation for a peculiar procedure ( or/and phenomenon ) would be a portion of our hereafter research work.

MATLABA® package version 7 from Mathworks, Natick, Massachusetts was employed for the ANN mold. The Levenberg-Marquardt optimisation algorithm was the back extension technique selected for developing the nervous web. The algorithm is the fastest back extension algorithm in MATLABA® tool chest. Hyperbolic tangent sigmoid and additive maps were the transportation maps for the hidden and the end product beds, severally. The public presentation of the ANN theoretical account was tested by 12 new and independent datasets. One hidden bed was selected in the ANN theoretical account. Harmonizing to the cosmopolitan estimate theory, one hidden bed with a sufficient figure of nerve cells can pattern any set of input/output informations to a sensible grade of truth ( Tambe et al. , 1996 ) .

Table 2 shows the public presentation of nervous web theoretical accounts with assorted Numberss of nerve cells in a individual hidden bed. The public presentation of the web was tested by ciphering R-squared, average absolute per centum mistake, and maximum/minimum absolute per centum mistakes for preparation ( including proof ) and proving datasets. The corresponding expression for the above statistical parametric quantities are presented in Appendix A. Generally, an R2 value larger than 0.9 shows a satisfactory public presentation for the proposed theoretical account ; while, an R2 extent in the scope of 0.8-0.9 is an index of a good public presentation and values lower than 0.8 indicate an unacceptable public presentation for the theoretical account suggested to foretell a peculiar variable. R2 was found higher than 0.998 for nervous webs with 4 and higher Numberss of nerve cells. High values for mean and maximal per centum mistakes were obtained when the figure of nerve cell was less than 4. Increasing the figure of nerve cells from 2 to 8 lowered the average per centum mistake by 93 % from 66.4 % to 4.5 % . The maximal and minimal absolute mistakes for the web with 8 nerve cells were 29 % and 0.01 % , severally. Increasing the figure of nerve cells from 8 to 10 increased the maximal absolute per centum mistake of the preparation and proving stages by 30 % and 45 % , severally. Higher Numberss of nerve cells were avoided in order to forestall overtraining of the ANN theoretical account ( Omidbakhsh et al, 2010 ) .

Table 2: Performance of ANN with assorted Numberss of nerve cells in one hidden bed

Figure 3 shows scatter secret plans of the predicted ( ANN ) BSA osmotic force per unit area versus the experimental information ( left ) and the per centum mistake for the preparation ( including proof ) and proving informations points. Despite the limited handiness of experimental informations, the ANN theoretical account ( 3:8:1 ) was capable of foretelling BSA osmotic force per unit area with a average absolute per centum mistake of about 5 % for both preparation and proving stages.

## Colloidal Interaction Model

## Zeta Potential of BSA

Zeta potency of BSA was estimated using the charge ordinance theoretical account for a pH scope of 4-10 and an ionic strength scope of 0.01-1M ; consequences are shown in Figure 4. Surface charge and zeta potency of BSA were calculated based on the ionisation reaction of aminic acids. Type and figure of aminic acids take parting in the ionisation reaction, pH, and ionic strength were required in the development of the charge ordinance theoretical account. The amino acerb sequence of BSA indicated the figure of each amino acid in the protein molecule ( Bowen and Williams, 1996a ) . Sodium chloride was assumed to be the lone salt in the solution. BSA surface carried a positive net charge at pH values lower than its isoelectric point and a negative net charge at pH values higher than the isoelectric point. At the isoelectric point, the surface net charge and accordingly zeta potency were zero. The estimated isoelectric point for BSA varied between 4.4 and 5.2, depending on the ionic strength of the solution. The experimental isoelectric point for BSA in 0.15M NaCl solution was reported to be 4.72 ( Vilker et al. 1984 ) which is in good understanding with the value estimated in this survey ( a‰? 4.76 ) . Results showed that zeta potency was strongly dominated by the consequence of pH, while ionic strength had a lower influence on the magnitude of zeta potency. Consequence of pH on the zeta potency was found more important at low values of ionic strength. The predicted zeta potency of BSA at ionic strength of 0.03M in Na chloride solution was compared with the experimental information ( Bowen and Williams, 1996a ) and acceptable understanding was observed.

## Osmotic Pressure

Osmotic force per unit area of BSA as a map of protein concentration was calculated at assorted pH and ionic strength values based on the colloidal interaction theoretical account. Figure 5 shows the consequences obtained in this stage of work. Electrostatic, London-van der Waals, and hydration forces every bit good as entropic force per unit area were considered in the colloidal interaction theoretical account. At pH near to the isoelectric point of BSA, the net charge on the protein was little and hence, electrostatic repulsive force and osmotic force per unit area were low. Osmotic force per unit area increased as pH diverged from the isoelectric point. Increase in the ionic strength, nevertheless, reversed the consequence by screening charges, doing molecular contraction and thereby diminishing the osmotic force per unit area. At an arbitrary BSA concentration of 300 g/L and ionic strengths of 0.03 and 0.15M, osmotic force per unit area decreased by about 70 % and 35 % , severally, when pH dropped from 7.4 to 5.4. It can be concluded that osmotic force per unit area becomes more sensitive to the ionic strength as pH diverges from the isoelectric point of BSA. At a BSA concentration of 300 g/L and pH of 5.4, osmotic force per unit area decreased by about 60 % when ionic strength increased from 0.03 to 0.15M. At the same BSA concentration and a pH value of 9, increasing the ionic strength from 0.03 to 0.15M resulted in the osmotic force per unit area decrease by about 80 % .

Performance of the colloidal interaction theoretical account was evaluated by plotting predicted osmotic force per unit area of BSA versus experimental informations, as shown in Figure 6. The developed colloidal interaction theoretical account predicted the osmotic force per unit area of BSA with an R2 and intend absolute per centum mistake of 0.954 and 29 % , severally, without any adjustable parametric quantity. The colloidal theoretical account experienced lower truth at pH 4.5 due to over anticipation of BSA osmotic force per unit area. The average absolute per centum mistake was obtained 14 % , 8 % , and 68 % at pH 7.4, 5.4, and 4.5, severally. The maximal per centum mistake of 120 % which exhibited a big mistake was observed at pH 4.5 and ionic strength 0.15M when protein concentration was higher than 300 g/L. The values of maximal absolute per centum mistake were 34 % and 22 % at pH 7.4 and 5.4, severally. Scatter secret plan of per centum mistake versus protein concentration ( Figure 6 ( C ) ) and versus ionic strength ( informations non shown ) followed no specific tendency.

## ANN versus Colloidal Interaction Model

Table 3 lists the osmotic force per unit area informations obtained by the ANN and colloidal interaction theoretical account and the per centum mistake for each information point. The consequences confirms that the osmotic force per unit area calculated by the nervous web theoretical account ( 3:8:1 ) is more accurate than the osmotic force per unit area calculated by the colloidal interaction theoretical account.

Since the ANN theoretical account was trained for limited scopes of pH, ionic strength and BSA concentration, it was indispensable to measure the interpolation/extrapolation power of the nervous web theoretical account. Nervous web anticipations at new pH, ionic strength, and protein concentration were compared merely against colloidal interaction theoretical account anticipations because experimental informations of BSA osmotic force per unit area was non available for all values of pH, ionic strength, and BSA concentration. The optimal nervous web theoretical account dwelling of one hidden bed with 8 nerve cells ( 3:8:1 ) was selected for the comparing intents. The absolute comparative difference was calculated at each information point, as depicted in Figure 7. A comparing was made at ionic strengths of 0.03, 0.15 and 1 grinder when the pH varies between 4.5 and 7.4.

Absolute comparative difference contour secret plans in Figure 7 showed maximal extremums at pH equal to 7. Absolute comparative difference was found every bit high as 800 % at ionic strength of 0.03M. The maximal absolute comparative difference of 1A-103 % was observed at pH 7 and ionic strength of 0.1 and 0.15M. Although accurate osmotic force per unit area values were predicted at pH 4.5, 5.4, and 7.4 by an ANN theoretical account dwelling of one hidden bed with 8 nerve cells, the ANN theoretical account ( 3:8:1 ) failed to extrapolate. Negative osmotic force per unit area values were predicted by the nervous web ( 3:8:1 ) at pH 7 when protein concentration was below 300 g/L. The insertion failure by the ANN theoretical account ( 3:8:1 ) was due to either limited available experimental informations or web overtraining ( Omidbakhsh et al. , 2010 ) . The overtraining of the nervous web was assessed by mapping the absolute comparative difference of the ANN theoretical accounts at different nerve cell Numberss. A nerve cell web with 4 nerve cells ( 3:4:1 ) appeared to be an optimal construction with the lowest absolute comparative difference. Absolute comparative difference contour for a nervous web dwelling of one hidden bed with 4 nerve cells was plotted in Figure 8.

A nerve cell web with 4 nerve cells ( 3:4:1 ) predicted the osmotic force per unit area with a maximal absolute comparative difference of 192 % , 186 % , and 182 % at ionic strengths of 0.03, 0.1 and 0.15M. ANN theoretical account ( 3:4:1 ) was able to foretell the osmotic force per unit area of BSA with a average absolute comparative difference of 41 % , 30 % , and 25 % . It was observed that the maximal comparative difference extremums fell in a part where the BSA concentration was lower than about 100 g/L. The maximal absolute comparative difference between the ANN theoretical account ( 3:4:1 ) anticipations and the colloidal interaction theoretical account anticipations for a pH scope of 4.5-7.4, ionic strength scope of 0.03-0.15M, and BSA concentration scope of 100-450 g/L was obtained less than 80 % . It is deserving adverting once more that colloidal interaction theoretical account anticipation was subjected to a maximal absolute per centum mistake of 120 % ( compared against experimental informations ) at pH 4.5, ionic strength 0.15M, and protein concentrations & gt ; 300 g/L. Osmotic force per unit area values predicted by the ANN theoretical account ( 3:4:1 ) were compared against the experimental information, ANN ( 3:8:1 ) consequences, and the colloidal interaction theoretical account anticipations, as tabulated in Table 3. Although ANN ( 3:4:1 ) had lower preciseness compared to the ANN ( 3:8:1 ) in foretelling experimental informations, ANN theoretical account ( 3:4:1 ) interpolated the BSA osmotic force per unit area with much less percent difference.

## 4.4 Relative Consequence of Input Variables

The part of each input variable in the nervous web on the BSA osmotic force per unit area was determined by a methodological analysis proposed by Garson for partitioning the neural connexion weights ( Garson, 1991 ) . The comparative importance of input parametric quantities, RI, was calculated utilizing the input and end product connexion weight ( Equation ( 13 ) ) . The higher correlativity between any input variable and the end product variable indicated greater significance of the variable on the magnitude of the dependent parametric quantity.

where New Hampshire is the figure of concealed nerve cells, nv the figure of input nerve cells, ivj the absolute value of input connexion weights, and Oj is the absolute value of connexion weights between the hidden and end product beds. Figure 9 illustrates the comparative importance of ionic strength, pH, and BSA concentration on the osmotic force per unit area. Ionic strength and pH were the most of import parametric quantities impacting the osmotic force per unit area.

## 5. Decisions

Osmotic force per unit area of BSA was predicted using unreal nervous web and colloidal interaction theoretical accounts. The nervous web consisted of an input bed with 3 nodes ( pH, ionic strength and BSA concentration ) , a concealed bed, and an end product bed ( osmotic force per unit area ) . The nervous web was trained utilizing the Levenberg-Marquardt optimisation algorithm. In the colloidal interaction theoretical account, particle-particle interactions such as electrostatic, London-van der Waals, and hydration forces along with entropy force per unit area were considered. Physicochemical belongingss of the BSA solution were considered in the colloidal theoretical account in order to gauge the osmotic force per unit area with no adjustable parametric quantities. The following chief decisions may be drawn from the consequences presented in this survey:

Prediction of BSA osmotic force per unit area was made possible utilizing ANN and the colloidal interaction theoretical account. The anticipation public presentation of the proposed nervous web was better than that of the colloidal interaction theoretical account. The colloidal theoretical account experienced low truth at pH 4.5 due to over anticipation of BSA osmotic force per unit area. The maximal absolute per centum mistakes of 120 % and 29 % were observed for the colloidal interaction theoretical account and ANN ( 3:8:1 ) , severally.

The cardinal facet of accurate anticipation of BSA osmotic force per unit area is the construction of the nervous web theoretical account. The optimal constellation for the ANN theoretical account within the scopes of input variables included 8 nerve cells in one hidden bed which was determined utilizing a test and mistake technique. The optimal ANN ( 3:8:1 ) predicted the BSA osmotic force per unit area with a average absolute per centum mistake of 4.5 % .

The insertion public presentation of the ANN theoretical account was investigated by ciphering the absolute comparative difference of the consequences obtained from the ANN theoretical account and the colloidal theoretical account. The ANN ( 3:8:1 ) failed to extrapolate due to either limited available experimental informations or web overtraining. A considerable betterment in the insertion public presentation of the ANN theoretical account was achieved by cut downing the figure of nerve cells from 8 to 4.

The comparative importance of the input variables was calculated utilizing the connexion weight partitioning method. Consequences showed that ionic strength and pH have the most effects on the magnitude of the osmotic force per unit area.

Although the proposed ANN theoretical account accurately predicts the BSA osmotic force per unit area, the theoretical account suffers from early convergence by the degeneration of several dimensions ; even if no local optima exist for the instances considered with this theoretical account. Hence, an efficient evolutionary algorithm is required to be combined with ANN which is a portion of our hereafter survey.

## Appendix A

R-Squared ( R2 ) , average absolute per centum mistake, and maximum/minimum absolute per centum mistakes are the statistical parametric quantities to prove the truth of the ANN theoretical account compared with the colloidal interaction theoretical account and experimental informations. The equations to calculate the above parametric quantities and besides the corresponding description are as follows:

R2 is a statistic parametric quantity which gives information on goodness of tantrum in a theoretical account. In arrested development analysis or adjustment of a theoretical account, the R2 coefficient is a statistical step of how good the theoretical account line approximates the existent information points. An R2 of 1.0 indicates the theoretical account line absolutely fits the information. The undermentioned equation shows the mathematical definition of R2 ( Montgomery and Runger, 2006 ; Montgomery, 2008 ) :

where M and P are the mensural and predicted osmotic force per unit area values, severally. represents the norm of the measured osmotic force per unit area informations.

Mean absolute per centum mistake ( MAPE ) is the step of truth in a fitted clip series value in statistics ( % ) and is defined as ( Montgomery and Runger, 2006 ; Montgomery, 2008 ) :

The Mean Squared Error ( MSE ) is a step of how close a fitted line or developed theoretical account is to data points. The smaller the Mean Squared Error the closer the tantrum ( or theoretical account ) is to the existent information. The MSE has the units squared of whatever is plotted on the perpendicular axis. MSE is described by the undermentioned relationship ( Montgomery and Runger, 2006 ; Montgomery, 2008 ) :

where N is the figure of samples.

Other two public presentation steps that were used in this paper to measure the effectivity of the preparation and proving informations include Minimum Absolute Error ( MIAE ) and Maximum Absolute Error ( MAAE ) as follows ( Montgomery and Runger, 2006 ; Montgomery, 2008 ) :