- Email: [email protected]
Mixed integer optimization models for the Synthesis of Protein Purification Processes with product loss
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Published by Elsevier Science B.V.
373
Mixed integer optimization models for the Synthesis of Protein Purification Processes with product loss Elsa Vasquez-Alvarez, Jose M. Pinto' Department of Chemical Engineering, University of Sao Paulo Av. Prof. Luciano Gualberto t. 3 n. 380, Sao Paulo, SP, 05508-900 Brazil
Abstract The objective of this work is to develop a mixed integer linear programming (MILP) model for the synthesis of protein purification processes. Mathematical models for each chromatographic technique rely on physicochemical data on the protein mixture, which contains the desired product and provide information on its potential purification. In previous works, MILP models assumed the complete recovery of the desired protein. The present model incorporates losses in the target protein along the purification process, in order to evaluate the trade-off between product by purity and quantity. A formulation that is based on a convex hull representation is proposed to calculate the minimum number of steps from a set of chromatographic techniques that must achieve a specified purity level as well as the amount of product recovered. Model linearity is achieved by assuming that the product is recovered in discrete percentages. The methodology is validated in examples with experimental data.
1. Introduction Many pharmaceutical products are proteins or polypeptides. These biotechnological products can be obtained from nature by extraction or produced by microorganisms genetically modified, namely recombinant proteins. In both cases, separation and purification of the desired protein are usually the most difficult stages in the whole process, and such stages may account for up to 60% of total cost (Lienqueo and Asenjo, 2000). Depending on the degree of complexity of the mixtures that result from bioreactions, several recovery and purification operations may be necessary to isolate the desired product. The most important operations include chromatographic techniques that are critical for therapeutical products such as vaccines and antibiotics that require very high purity levels (98 - 99.9%). One of the main challenges in the synthesis of downstream purification stages is the appropriate selection and sequencing of chromatographic steps (Larsson et ai, 1997). Therefore, optimization methods (Vasquez-Alvarez et al., 2001) as well as expert systems (Bryant and Rowe, 1998) are useful tools for the design and synthesis of protein purification processes. Steffens et ai (2000) developed a synthesis technique for generating optimal downstream processing flowsheets for ^Author to whom all correspondence should be addressed. E-mail: [email protected]. Financial support from PADCT/CNPq under grant 62.0239/97 - QEQ, from ANTORCHAS and VITAE (Coop. Programs Argentina - Brasil - Chile) under grants A-13668/1-9 and B-11487/10B006.
374 biotechnological processes. The technique integrates the idea of screening units via physical property information into an imphcit enumeration synthesis tool. Mathematical programming approaches for process synthesis rely on the representation of algebraic equations with discrete variables. In previous works, Vasquez-Alvarez et al. (2001) and Vasquez-Alvarez and Pinto (2001) developed mixed-integer linear optimization models that implicitly assume the complete recovery of the desired protein. In the present work, the objective is to incorporate losses in the target protein along the purification process, in order to evaluate the trade-off between product quality given by purity and quantity.
2. Problem Description Consider a complex protein mixture that must be purified by chromatographic techniques. The degree of separation depends on the protein partition differential between the stationary and mobile phases. Information on physicochemical properties can be used for the target and contaminant proteins and each chromatographic technique is able to perform the separation of the mixture by exploiting a specific physicochemical property, such as surface charge as a function of pH, surface hydrophobicity, molecular weight. For instance, ion exchange chromatography separates proteins based on their difference in charge. The charge of a protein depends on pH according to the titration curve. Ion exchange can use small differences in charge that yield a very high resolution and hence it is an extremely efficient operation to separate proteins. Losses of target protein along the purification process are possible, and therefore we must evaluate the trade-off between product quality (given by purity) and quantity. In other words, the higher the purity achieved within each step, the smaller the product yield. In this sense, decisions involve the selection of techniques and their order as well as the percentage of product recovered.
3. Mathematical Model Mixed-integer linear optimization models for the synthesis of purification bioprocesses were developed in previous work (Vasquez-Alvarez, et al., 2001). In previous models, implicit was the assumption that product recovery was complete. Consequently, the major decisions concerned selection and ordering of chromatographic operations. In the present case, the model must account for target protein losses along the purification process. In order to keep the optimization model linear, it is assumed that product must be recovered in discrete percentages. 3.1 Model of a chromatographic technique The approach for each chromatographic technique is to approximate chromatograms by isosceles triangles. Moreover, physicochemical property data for the target protein as well as for the major contaminants are required. It is assumed that the peaks in chromatograms have constant shapes and that the one on the left refers to the product and the other to the contaminant protein. In Figure 1, the shaded areas represent product that is removed with the contaminants for a given
375 discrete recovery level (given by the index /). On the other hand, the dark-shaded areas (with base Bipi) represent the amount of contaminant p that remains in the mixture (with the product) after applying chromatographic technique /. It is important to note that five situations may arise, depending to the relative position of the triangles. The first case corresponds to an almost complete overlap between the triangles; the other extreme occurs when both triangles are completely apart; finally, the remaining cases are shown in Figures 1(a), 1(b) and 1(c). In such figures, the amount of lost product can be determined from the ratio of the light-shaded area and the total area of the product chromatogram. Similarly, the amount of contaminant that remains in the mixture is calculated from the ratio of the dark-shaded and the total area. ^^>
DF,p
Figure 1. Peak representations in a chromatogram (a), (b) and (c) The mathematical correlations applied for each chromatographic technique are given in Table 1 for protein p at chromatographic step / with discrete recovery level /. The ratio of proteins (p and dp) that remain in the mixture after and before chromatographic technique / at separation level / is denoted by CFipj. The relationships expressed in Table 1 represent graphical approximations of the chromatograms for two different proteins. As a result, a fraction of proteins is admitted not to separate from the product (represented as 1.02 coefficients in Table 1). In Table 1, the first row indicates that purification is not carried out. In the following rows the purification degree increases (correspond to figures 1(a), 1(b) and (Ic)), up to the case of total separation {CFi,p = 0.02). The concentration factors CFip shown in Table 1 are introduced in the synthesis model that is described in the next section. 3.2 Synthesis model An optimization model that minimizes the total number of chromatographic steps for a given purity level is proposed. This model relies on a convex hull representation that is derived from the following general disjunction: / L V V
\/p
V
^k
k=\...K-l
(1)
Disjunction (1) contains /.L-f 1 elements for each order k. The first I.L terms model the selection of step / in order k at level /, whereas the last term models no step selection.
376 Table J. Mathematical relationships for chromatographic techniques Deviation Factor
Base Bj^pi
0 < DF,n < ^ '' 10
B,pj = Ci-DF,_p
-^-DF,,p<
0< ^
B,,pj
B,,pj<-^-DF,,p
(fig 2a)
(figs. 2a and 2b)
^
M a s s reduction of protein p \/p
CF,,,,,= 1
CF,,pj=\.02
[\-2{
CF,,p,,= \.02 [2
i^i-DFi^p-Bi,p,l)
)]
(Dfi^p+Bj^pj)
i^i-^i CF,,,,= 1 . 0 2 [ l - ( 2 - ^
P-^P
p=dp
pi) ^^^i-)]
VpM?
of 0
Vp^p
- ^ )
(fig- 2b) CF,^,/= 1.02 [1-2 ( -±
^^
^^
)]
P=dp
0 < 5,p./< —
(fig. 2b)
CF,,,,/=1.02(2
DF,,p>ai
B,,.i = 0
- ^ )
Vp^p Vp
CF,,„/= 0.02
V/t
(2a)
v/
(2bj
VA:
(3a)
VA:
(3bj
V/:,/t'
(3c)
V/t,A:'>)t
(3d)
k I
(3e) k
^p,2=Y.Y.^^i,p,i\xi'^p,\ i I ^pM\
=^I.^^i,p,l-f^lp,kJ
(4a)
Vp, k=2...K-\
(4b)
Mp, k=2...K-l
(4c)
yi,p,l.k=2...K-i
(4d)
yp, k=2...K-\
(4e)
+'^M
2 P,k=im^i,p,k,l-^^l.p.k i I \pM
mp,,
Vp
-i/(l-Z^)
k=\...K-\
(5)
377 ^^,^+1 ^/^-Z^pM-^.d-Z,)
k=\...K-\
(6)
p
Xa./e{0,l}
VU,/
m,,,,Z^ m^,^,;,;, m^^^, a , > 0 V/, p,/:, /
(7)
Constraint (2a) indicates that at most one step / may be chosen in order k. Slack variable a^t is activated if no steps are selected in order k. Constraint (2b) imposes that step / is selected at most once in the sequence and (3a) states that steps are assigned in increasing order. Constraints (3b)-(3e) define the last step of the sequence, denoted by Zk Constraint set (4) relates subsequent steps and is generated from disjunction (1). Constraints (5) and (6) enforce purity and yield specifications, respectively. Objective function (8) that selects a minimum sequence for a given purity as well as yield is as follows:
MinS=Y^Y.Y.Ku=J.k.Z, i
k
I
(8)
k
Alternatively, profit can be maximized by simply taking into account the revenue from product sales and operating costs of the chromatographic columns with available data.
4. Computational Results The software GAMS/CPLEX 7.0 (Brooke et al., 1998) was used to implement the MILP model and to generate its solution. Two different examples of increasing size are solved, which correspond to the first two presented in Vasquez-Alvarez et al. (2001). In Example 1, taken from Lienqueo et al. (1998), we consider the purificafion of a mixture containing four proteins, all in equal concentration. Their physicochemical properties as well as the initial protein concentration of the mixture are shown in Vasquez-Alvarez et al. (2001). The required purity level for pi is 98%. If no product loss is considered, results are the same as those of model (M^) by Vasquez-Alvarez et al. (2001), which comprise three steps and 99.8% final purity. Nevertheless, if 4% loss of product is accepted, only two steps are necessary and 99.9% final purity is achieved. In Example 2, we consider the purification of (3-1,3 glucanase (8.3% initial concentrafion) that must be separated from eight contaminants; twenty-two chromatographic techniques are available (Lienqueo et al., 1999). Consider Case a that corresponds to 94% purity and 100% recovery of P-1,3 glucanase and Case b given by 99% purity with 6% of product losses. In Case a, results are the same as in VasquezAlvarez et al (2001), and is shown in Fig. 2 (6 steps and 94.8% for final purity). For Case b, three techniques are employed and 99.7% final maximum purity is achieved, as is shown in Fig. 3. Statistical data for both examples are given in Table 2. Table 2. Summary of statistical data for examples 1 and 2 Example 1 2
L
fr(%)
1 3 1 4
100.0 96.0 100.0 94.0
Integer . ^1 vanables 168 456 288 1080
Continuous • i_i variables 618 1674 2378 8912
^ , ., Constraints
Nodes
874 1930 2699 9233
20 0 623 972
CPU times (s) 1.46 1.87 138.6 1333
378 Sl£il
"r? 0.083
0.925
0811
I
0.948
map 1 =0.62 Imap 1 =0.62|
Imdf i=0.62|
lmapo=0.62|
VJ Protein Mixture
Hydrophobic Interaction
Anion Exchange pH 8.5
Anion Exchange pH 6.0
Anion Exchange pH 5.0
Anion Exchange pH 6.5
Cation Exchange pH 7.0
Product
Fig 2. Optimal results for example 2 - Case a Sl£fi
0.083
0351
[map.o=0.62
Protein Mixture
0867
Anion Exchange pH 5 0
0.997
i
idM =0 608
Anion Exchange pH 7.0
Hydrophobic Interaction
D
Product
Fig 3. Optimal results for example 2 - Case b
5. Conclusions This paper presented the development of an MILP for the synthesis of chromatographic steps for the purification of protein mixtures considering product losses. Results indicate that a systematic selection and sequencing of chromatographic steps may be obtained by the appropriate balance between yield and purity level. Notation Bixi CFi^pj DFip Dp Fp Fr k Kdip /
width of contaminant peak, remains in mixture cone, factor of cont. p after step / in level / deviation factor for protein p in chrom. step / desired protein (product) specified purity level of dp specified yield level of dp order in the sequence (/: = \,...K) retention time of protein p in technique / level of target protein losses
nipk p S U Zk ttjt X,xi (7,
mass of p after technique in order k protein (product + contaminants) objective function variable upper bound on protein mass binary variable that indicates if order k is last slack variable relative to order k boolean variable for selecting technique / in order k at level / of product loss Width of chromatographic peak in step /
References Brooke A., D. Kendrick, A. Meeraus, and R. Raman, 1998, GAMS -A user's guide, The Scientific Press, Redwood City, USA. Bryant, C.H., and R.C. Rowe, 1998, Trac - Trends in Analytical Chemistry, 17, 18. Larsson, G., S.B. Jorgensen, M.N. Pons, B. Sonnleitner, A. Tijsterman and N. Tichener-Hooker, 1997, J. Biotechnol.,59, 3. Lienqueo, M.E., J.C. Salgado and J.A. Asenjo, 1998, Comp. Appl. Biot. - CAB7, 321, Osaka, Japan. Lienqueo, M.E., J.C Salgado and J.A. Asenjo, 1999, J. Chem. Technol. Biot., 74, 293. Lienqueo, M.E., and J.A. Asenjo, 2000, Comput. Chem. Eng., 24, 2339. Steffens M.A., E.S. Fraga and I.D.L. Bogle, 2000, Biotechnol. and Bioeng., 68, 2, 218. Vasquez-Alvarez E., M.E. Lienqueo and J.M. Pinto, 2001, Biotechnol Prog., 17, 685. Vasquez-Alvarez E. and J.M. Pinto, 2001, 579, European Symposium on Computer Aided Process Engng - 11, Eds. R. Gani and S.B. Jorgensen, Elsevier, Amsterdam.
373
Mixed integer optimization models for the Synthesis of Protein Purification Processes with product loss Elsa Vasquez-Alvarez, Jose M. Pinto' Department of Chemical Engineering, University of Sao Paulo Av. Prof. Luciano Gualberto t. 3 n. 380, Sao Paulo, SP, 05508-900 Brazil
Abstract The objective of this work is to develop a mixed integer linear programming (MILP) model for the synthesis of protein purification processes. Mathematical models for each chromatographic technique rely on physicochemical data on the protein mixture, which contains the desired product and provide information on its potential purification. In previous works, MILP models assumed the complete recovery of the desired protein. The present model incorporates losses in the target protein along the purification process, in order to evaluate the trade-off between product by purity and quantity. A formulation that is based on a convex hull representation is proposed to calculate the minimum number of steps from a set of chromatographic techniques that must achieve a specified purity level as well as the amount of product recovered. Model linearity is achieved by assuming that the product is recovered in discrete percentages. The methodology is validated in examples with experimental data.
1. Introduction Many pharmaceutical products are proteins or polypeptides. These biotechnological products can be obtained from nature by extraction or produced by microorganisms genetically modified, namely recombinant proteins. In both cases, separation and purification of the desired protein are usually the most difficult stages in the whole process, and such stages may account for up to 60% of total cost (Lienqueo and Asenjo, 2000). Depending on the degree of complexity of the mixtures that result from bioreactions, several recovery and purification operations may be necessary to isolate the desired product. The most important operations include chromatographic techniques that are critical for therapeutical products such as vaccines and antibiotics that require very high purity levels (98 - 99.9%). One of the main challenges in the synthesis of downstream purification stages is the appropriate selection and sequencing of chromatographic steps (Larsson et ai, 1997). Therefore, optimization methods (Vasquez-Alvarez et al., 2001) as well as expert systems (Bryant and Rowe, 1998) are useful tools for the design and synthesis of protein purification processes. Steffens et ai (2000) developed a synthesis technique for generating optimal downstream processing flowsheets for ^Author to whom all correspondence should be addressed. E-mail: [email protected]. Financial support from PADCT/CNPq under grant 62.0239/97 - QEQ, from ANTORCHAS and VITAE (Coop. Programs Argentina - Brasil - Chile) under grants A-13668/1-9 and B-11487/10B006.
374 biotechnological processes. The technique integrates the idea of screening units via physical property information into an imphcit enumeration synthesis tool. Mathematical programming approaches for process synthesis rely on the representation of algebraic equations with discrete variables. In previous works, Vasquez-Alvarez et al. (2001) and Vasquez-Alvarez and Pinto (2001) developed mixed-integer linear optimization models that implicitly assume the complete recovery of the desired protein. In the present work, the objective is to incorporate losses in the target protein along the purification process, in order to evaluate the trade-off between product quality given by purity and quantity.
2. Problem Description Consider a complex protein mixture that must be purified by chromatographic techniques. The degree of separation depends on the protein partition differential between the stationary and mobile phases. Information on physicochemical properties can be used for the target and contaminant proteins and each chromatographic technique is able to perform the separation of the mixture by exploiting a specific physicochemical property, such as surface charge as a function of pH, surface hydrophobicity, molecular weight. For instance, ion exchange chromatography separates proteins based on their difference in charge. The charge of a protein depends on pH according to the titration curve. Ion exchange can use small differences in charge that yield a very high resolution and hence it is an extremely efficient operation to separate proteins. Losses of target protein along the purification process are possible, and therefore we must evaluate the trade-off between product quality (given by purity) and quantity. In other words, the higher the purity achieved within each step, the smaller the product yield. In this sense, decisions involve the selection of techniques and their order as well as the percentage of product recovered.
3. Mathematical Model Mixed-integer linear optimization models for the synthesis of purification bioprocesses were developed in previous work (Vasquez-Alvarez, et al., 2001). In previous models, implicit was the assumption that product recovery was complete. Consequently, the major decisions concerned selection and ordering of chromatographic operations. In the present case, the model must account for target protein losses along the purification process. In order to keep the optimization model linear, it is assumed that product must be recovered in discrete percentages. 3.1 Model of a chromatographic technique The approach for each chromatographic technique is to approximate chromatograms by isosceles triangles. Moreover, physicochemical property data for the target protein as well as for the major contaminants are required. It is assumed that the peaks in chromatograms have constant shapes and that the one on the left refers to the product and the other to the contaminant protein. In Figure 1, the shaded areas represent product that is removed with the contaminants for a given
375 discrete recovery level (given by the index /). On the other hand, the dark-shaded areas (with base Bipi) represent the amount of contaminant p that remains in the mixture (with the product) after applying chromatographic technique /. It is important to note that five situations may arise, depending to the relative position of the triangles. The first case corresponds to an almost complete overlap between the triangles; the other extreme occurs when both triangles are completely apart; finally, the remaining cases are shown in Figures 1(a), 1(b) and 1(c). In such figures, the amount of lost product can be determined from the ratio of the light-shaded area and the total area of the product chromatogram. Similarly, the amount of contaminant that remains in the mixture is calculated from the ratio of the dark-shaded and the total area. ^^>
DF,p
Figure 1. Peak representations in a chromatogram (a), (b) and (c) The mathematical correlations applied for each chromatographic technique are given in Table 1 for protein p at chromatographic step / with discrete recovery level /. The ratio of proteins (p and dp) that remain in the mixture after and before chromatographic technique / at separation level / is denoted by CFipj. The relationships expressed in Table 1 represent graphical approximations of the chromatograms for two different proteins. As a result, a fraction of proteins is admitted not to separate from the product (represented as 1.02 coefficients in Table 1). In Table 1, the first row indicates that purification is not carried out. In the following rows the purification degree increases (correspond to figures 1(a), 1(b) and (Ic)), up to the case of total separation {CFi,p = 0.02). The concentration factors CFip shown in Table 1 are introduced in the synthesis model that is described in the next section. 3.2 Synthesis model An optimization model that minimizes the total number of chromatographic steps for a given purity level is proposed. This model relies on a convex hull representation that is derived from the following general disjunction: / L V V
\/p
V
^k
k=\...K-l
(1)
Disjunction (1) contains /.L-f 1 elements for each order k. The first I.L terms model the selection of step / in order k at level /, whereas the last term models no step selection.
376 Table J. Mathematical relationships for chromatographic techniques Deviation Factor
Base Bj^pi
0 < DF,n < ^ '' 10
B,pj = Ci-DF,_p
-^-DF,,p<
0< ^
B,,pj
B,,pj<-^-DF,,p
(fig 2a)
(figs. 2a and 2b)
^
M a s s reduction of protein p \/p
CF,,,,,= 1
CF,,pj=\.02
[\-2{
CF,,p,,= \.02 [2
i^i-DFi^p-Bi,p,l)
)]
(Dfi^p+Bj^pj)
i^i-^i CF,,,,= 1 . 0 2 [ l - ( 2 - ^
P-^P
p=dp
pi) ^^^i-)]
VpM?
of 0
Vp^p
- ^ )
(fig- 2b) CF,^,/= 1.02 [1-2 ( -±
^^
^^
)]
P=dp
0 < 5,p./< —
(fig. 2b)
CF,,,,/=1.02(2
DF,,p>ai
B,,.i = 0
- ^ )
Vp^p Vp
CF,,„/= 0.02
V/t
(2a)
v/
(2bj
VA:
(3a)
VA:
(3bj
V/:,/t'
(3c)
V/t,A:'>)t
(3d)
k I
(3e) k
^p,2=Y.Y.^^i,p,i\xi'^p,\ i I ^pM\
=^I.^^i,p,l-f^lp,kJ
(4a)
Vp, k=2...K-\
(4b)
Mp, k=2...K-l
(4c)
yi,p,l.k=2...K-i
(4d)
yp, k=2...K-\
(4e)
+'^M
2 P,k=im^i,p,k,l-^^l.p.k i I \pM
mp,,
Vp
-i/(l-Z^)
k=\...K-\
(5)
377 ^^,^+1 ^/^-Z^pM-^.d-Z,)
k=\...K-\
(6)
p
Xa./e{0,l}
VU,/
m,,,,Z^ m^,^,;,;, m^^^, a , > 0 V/, p,/:, /
(7)
Constraint (2a) indicates that at most one step / may be chosen in order k. Slack variable a^t is activated if no steps are selected in order k. Constraint (2b) imposes that step / is selected at most once in the sequence and (3a) states that steps are assigned in increasing order. Constraints (3b)-(3e) define the last step of the sequence, denoted by Zk Constraint set (4) relates subsequent steps and is generated from disjunction (1). Constraints (5) and (6) enforce purity and yield specifications, respectively. Objective function (8) that selects a minimum sequence for a given purity as well as yield is as follows:
MinS=Y^Y.Y.Ku=J.k.Z, i
k
I
(8)
k
Alternatively, profit can be maximized by simply taking into account the revenue from product sales and operating costs of the chromatographic columns with available data.
4. Computational Results The software GAMS/CPLEX 7.0 (Brooke et al., 1998) was used to implement the MILP model and to generate its solution. Two different examples of increasing size are solved, which correspond to the first two presented in Vasquez-Alvarez et al. (2001). In Example 1, taken from Lienqueo et al. (1998), we consider the purificafion of a mixture containing four proteins, all in equal concentration. Their physicochemical properties as well as the initial protein concentration of the mixture are shown in Vasquez-Alvarez et al. (2001). The required purity level for pi is 98%. If no product loss is considered, results are the same as those of model (M^) by Vasquez-Alvarez et al. (2001), which comprise three steps and 99.8% final purity. Nevertheless, if 4% loss of product is accepted, only two steps are necessary and 99.9% final purity is achieved. In Example 2, we consider the purification of (3-1,3 glucanase (8.3% initial concentrafion) that must be separated from eight contaminants; twenty-two chromatographic techniques are available (Lienqueo et al., 1999). Consider Case a that corresponds to 94% purity and 100% recovery of P-1,3 glucanase and Case b given by 99% purity with 6% of product losses. In Case a, results are the same as in VasquezAlvarez et al (2001), and is shown in Fig. 2 (6 steps and 94.8% for final purity). For Case b, three techniques are employed and 99.7% final maximum purity is achieved, as is shown in Fig. 3. Statistical data for both examples are given in Table 2. Table 2. Summary of statistical data for examples 1 and 2 Example 1 2
L
fr(%)
1 3 1 4
100.0 96.0 100.0 94.0
Integer . ^1 vanables 168 456 288 1080
Continuous • i_i variables 618 1674 2378 8912
^ , ., Constraints
Nodes
874 1930 2699 9233
20 0 623 972
CPU times (s) 1.46 1.87 138.6 1333
378 Sl£il
"r? 0.083
0.925
0811
I
0.948
map 1 =0.62 Imap 1 =0.62|
Imdf i=0.62|
lmapo=0.62|
VJ Protein Mixture
Hydrophobic Interaction
Anion Exchange pH 8.5
Anion Exchange pH 6.0
Anion Exchange pH 5.0
Anion Exchange pH 6.5
Cation Exchange pH 7.0
Product
Fig 2. Optimal results for example 2 - Case a Sl£fi
0.083
0351
[map.o=0.62
Protein Mixture
0867
Anion Exchange pH 5 0
0.997
i
idM =0 608
Anion Exchange pH 7.0
Hydrophobic Interaction
D
Product
Fig 3. Optimal results for example 2 - Case b
5. Conclusions This paper presented the development of an MILP for the synthesis of chromatographic steps for the purification of protein mixtures considering product losses. Results indicate that a systematic selection and sequencing of chromatographic steps may be obtained by the appropriate balance between yield and purity level. Notation Bixi CFi^pj DFip Dp Fp Fr k Kdip /
width of contaminant peak, remains in mixture cone, factor of cont. p after step / in level / deviation factor for protein p in chrom. step / desired protein (product) specified purity level of dp specified yield level of dp order in the sequence (/: = \,...K) retention time of protein p in technique / level of target protein losses
nipk p S U Zk ttjt X,xi (7,
mass of p after technique in order k protein (product + contaminants) objective function variable upper bound on protein mass binary variable that indicates if order k is last slack variable relative to order k boolean variable for selecting technique / in order k at level / of product loss Width of chromatographic peak in step /
References Brooke A., D. Kendrick, A. Meeraus, and R. Raman, 1998, GAMS -A user's guide, The Scientific Press, Redwood City, USA. Bryant, C.H., and R.C. Rowe, 1998, Trac - Trends in Analytical Chemistry, 17, 18. Larsson, G., S.B. Jorgensen, M.N. Pons, B. Sonnleitner, A. Tijsterman and N. Tichener-Hooker, 1997, J. Biotechnol.,59, 3. Lienqueo, M.E., J.C. Salgado and J.A. Asenjo, 1998, Comp. Appl. Biot. - CAB7, 321, Osaka, Japan. Lienqueo, M.E., J.C Salgado and J.A. Asenjo, 1999, J. Chem. Technol. Biot., 74, 293. Lienqueo, M.E., and J.A. Asenjo, 2000, Comput. Chem. Eng., 24, 2339. Steffens M.A., E.S. Fraga and I.D.L. Bogle, 2000, Biotechnol. and Bioeng., 68, 2, 218. Vasquez-Alvarez E., M.E. Lienqueo and J.M. Pinto, 2001, Biotechnol Prog., 17, 685. Vasquez-Alvarez E. and J.M. Pinto, 2001, 579, European Symposium on Computer Aided Process Engng - 11, Eds. R. Gani and S.B. Jorgensen, Elsevier, Amsterdam.