Biotechnol. Prog. 1991, 7, 72-75
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Application of Population Balance Model to the Loss of Hybridoma Antibody Productivity Gyun Min Lee, Amit Varma, and Bernhard 0. Palsson* Department of Chemical Engineering, Herbert H. Dow Building, University of Michigan, Ann Arbor, Michigan 48109
A simple dynamic model has been applied t o explain t h e population dynamics of monoclonal antibody (MAb) producing (producer) a n d nonproducing hybridoma cells (nonproducer) coexisting in culture. T h e events of mutation or loss of genes associated with antibody synthesis have been incorporated into t h e model t o account for t h e conversion of a producer t o a nonproducer. T h e model shows t h a t t h e cell population is not necessarily dominated by t h e nonproducer, a n d a steady balance of producer and nonproducer populations can be achieved. A nonproducer population is undesirable, and cultivation strategies t o maximize M A b production are suggested, taking into account t h e dynamics of a nonproducer population.
Introduction Loss of monoclonal antibody (MAb) productivity of hybridomas in long-term cultivation has become a major concern to those interested in the large-scale production of MAbs. Although the mechanisms involved in the loss of MAb productivity have not been fully understood, random mutations or loss of genes associated with antibody regulation, as well as antibody synthesis, have been suggested (GalfrB et al., 1980; Gardner et al., 1985; Hengartner et al., 1978). Irreversible loss of productivity provides evidence in support of this argument (GalfrB et al., 1980; Gardner et al., 1985; Lee and Palsson, 1990; Wilde and Milstein, 1980). The appearance of a nonproducer (NP) population in culture has been observed in several fashions by many investigators. Murine hybridoma cell lines HB8178 and AFP-27 lost MAb productivity in continuous culture using Dulbecco’s modified Eagle’s medium containing 10% horse serum (Frame and Hu, 1990). Murine hybridoma cell line 167.4G5.3 lost MAb productivity in 1.25% fetal bovine serum (FBS) supplemented medium over a time period of about 4 months, while the cells maintained MAb productivity in 5 OE serum supplemented medium for the same period (Ozturk and Palsson, 1990b). The MAb productivity of murine hybridoma cell line S3H5/?2bA was maintained in 1CT;. FBS supplemented medium, while it completely disappeared in IMDM-based low-protein medium (Lee and Palsson, 1990). Although the tendency for loss of MAb productivity is particular to each cell line, cells in low-serum medium may be more prone to lose MAb productivity than cells in high-serum medium. When a N P appears in culture, it does not always take over the whole culture (Gardner et al., 1985; Westerwoudt et al., 1984). Populations of producer (P)and N P may be balanced under certain culture conditions, depending on the relative growth rate of producing and nonproducing cells and the rate of loss of MAb productivity. With current hybridoma technology, it is difficult to make stable cell lines free of genetic instability. Therefore, it is important to find the conditions where a stable balance of producing and nonproducing cell populations can be obtained. Mathematical models for population balance have been
used to interpret data on the appearance of mutants in bacterial populations (Atwood, 1951; Moser, 1957; Rubinow, 1975). This modeling approach has been used to explain the segregative instability in plasmid containing bacterial populations (Imanaka and Aiba, 1981). The loss of antibody productivity can be described by a similar model. We have applied a dynamic population model to understand the effect of relative rates of growth and loss of MAb productivity on a P and N P population balance. On the basis of this model, we are able to suggest cultivation strategies for overcoming the loss of MAb productivity.
The Model In developing the model to describe a population balance between P and N P in long-term cultivation, we have made the following assumptions: (1)Loss of antibody productivity is due to mutation or loss of genetic material, and is realized following division, e.g. in a newborn cell that becomes a NP. (2) Growth characteristics of P and N P are uniform within their populations. (3) Extended periods of growth are considered, which may be obtained by subculturing cells repeatedly in the late exponential phase of growth. (4)Since cells are always in the exponential phase, cell death is neglected. (5) Loss of antibody productivity is irreversible. The cell population balance equations for such an exponentially growing culture are dxp/dt = / . L ~ x -, C Y X ,
(1)
dxn/dt = l n x n + axp
(2)
where xp is the viable cell density of P , xn is the viable cell density of NP, CY is the rate of loss of MAb productivity, pp is the specific growth rate of P, / . L ~is the specific growth rate of NP, and t is the cultivation time. The parameter a is defined as CY = a/(ln 2/ccP) (3) where a is the probability of loss of MAb productivity. If we define dimensionless variables X, = xp/xpo,X, = xn/xpo(where xpo is the initial cell density of P), and 7 =
8756-7938/91/3007-0072$02.50/00 1991 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1991, Vol. 7, No. 1
73
p p t and dimensionless parameters S = pn/ppand A = a/pP
= a / l n 2, eqs 1 and 2 can be written as
d X p / d r = (1- A)X, dX,/dT = SX,
(4)
+AX,
"T
Non-producers take over.
(5)
T h e variation in the dimensionless cell concentrations,
X, and X,, over time is obtained analytically by solving eqs 4 and 5 simultaneously: ~
~
e('-')' )
(6)
( =7
1
0.980
X,(T) = (AT
+ Xno)esr
if 1- A - S = 0
= [A/(1 - A - S)][e('-''' - eS']
+ XnoeS'
otherwise (7)
0.970
'
1;-6
. ...... 1;-5
. . ..... . . ...... . . ...... . '....... 1;-4
1;-3
where X,O is the initial cell density of NP.
Condition for a Balanced Population. Fraction of N P in the total population is
F = X,/(X, + X,) (8) When T goes to infinity (steady state), F can have two different values depending on the numerical value of 1 A - S:
F = lim [X,/(X, r--
+ X,)]
=1
if 1- A - S I O
= A / ( 1 - S ) if 1 - A - S
> 0 (9)
Figure 1shows the conditions for a balanced population. N P will take over the culture if 1- A - S I0, which is the case observed by many investigators (Frame and Hu, 1990; Heath et al., 1990; Lee and Palsson, 1990; Ozturk and Palsson, 1990b). Equation 9 confirms that a balanced population between P and N P , which has been observed by several investigators (Gardner et al., 1985;Westerwoudt et al., 1984), can also be obtained if 1 - A - S > 0.
Effect of Relative Growth Rates and Mutation Rates on a Population Balance. It is likely that the proportion of hybridomas exhibiting a loss of MAb production may be higher shortly after fusion, compared to when established as clone in culture (Westerwoudt et al., 1984). Mutation rates can vary by 3 orders of magnitude and are on the order of 10-2-10-5/generation (Clark et al., 1983; Cotton et al., 1973; Galfrg et al., 1980; Gardner e t al., 1985). As shown in Figure 1,the population balance in this range of mutation rate is so sensitive to relative growth rate that N P can take over the whole culture with a slight growth edge over P. The effect of relative growth rate and mutation rate on fraction of N P , using initial condition of X,, = 0, are shown in Figure 2. The generation number, n, is defined as
n = r/ln 2 (10) As shown in Figure 2, N P does not take over the whole culture when A S < 1. The fraction F increases in the beginning and remains fairly constant with future generations. As Gardner and colleagues (1985) claim, preferential outgrowth of N P is not a major factor in this condition and random mutation is responsible for the generation of N P (Gardner e t al., 1985). If the growth rate of N P is 10% lower than that of P with the A value in the range of and the population of N P in the culture is insignificant. Long-term cultivation can be achieved without significant loss of MAb productivity, and recloning is unnecessary for maintaining the cell line.
+
1;-2
1;-1
' A
Figure 1. Condition for a balanced producer and nonproducer population. The region above the curve where A + S C 1 corresponds to conditions where both producer and nonproducer
populations can coexist in a steady state. Below the curve nonproducers outgrow the producers and take over the culture.
However, if the growth rate of N P is 10 % higher than that of P with an A value of 10-3, F , the fraction of N P in the total population, is 0.1 within 30 generations and increases veryrapidlythereafter. The fraction F a t S = 1.2 increases more rapidly than a t S = 1.1 and is over 0.3 after 30 generations. Under these conditions, frequent recloning may be required to maintain P. Figure 3 shows that simulated result agrees well with Ozturk and Palsson's measurement of the fraction of nonproducers during long-term cultivation of hybridoma cells in 1.25% serum medium (Ozturk and Palsson, 1990b). Values of A and S, which were obtained by using the MINPACK software package with two-parameter estimation (Garbow et al., 1980), were found to be 1.7 X loe3 and 1.072, respectively. These numerical estimates for these two parameters are quite reasonable. The estimated value for A falls within the reported range of 10-2-10-5 cell-' generation-' (Clark et al., 1983; Cotton e t al., 1973; Galfr6 et al., 1980; Gardner et al., 1985). The value of S indicates a 7 7%faster growth rate for the nonproducing clone, which eventually outgrows the producer.
Discussion Several hybridoma cell lines have been reported to lose their MAb productivity a t the rate of 10-2-10-5 cell-' generation-l (Clark et al., 1983; Cotton et al., 1973; Galfr6 et al., 1980; Gardner et al., 1985). The appearance of N P is a major concern to those interested in large-scale production of MAbs. For the economical production of MAb, the fraction of N P should be maintained as low as possible or MAb production should be completed before N P becomes dominant in the culture. When N P appears in the culture, it is expected to take over the whole culture (Goding, 1980; Lemke, 1979). However, in many cases, it has been observed that P and N P are balanced in the culture (Gardner et al., 1985; Westerwoudt e t al., 1984). Our model shows that a balanced population can be achieved a t particular conditions of mutation rate and reltive growth rates ( A S < 1). On the basis of the simple dynamic model presented here, we can suggest cultivation strategies for MAb production. If 1 - A - S > 0, long-term cultivation of producing cells can be achieved and recloning for maintaining the producer is unnecessary. I t is therefore
+
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Biotechnol. Prog., 1991, Vol. 7, No. 1
(A) A=10"
(B) A = 1 0 ' 3
1
1
"O
8r1.2
L
0.e
0
20
60
40
60
/ 100
n
n 1.0-
0.6
L 0.4
0.2 0.2
I
-
-
0.0 0
/
0.8
-
0.6
-
LL 0.4
--/
1 .o
/
/
0.2
(D) A=lO"
1 .os
0.0 20
40
60
80
100
0
20
40
60
80
100
n n Figure 2. Effect of relative growth rate and mutation rate on a population balance. A stable population balance is seen for S = 0.9. All other simulations show an eventual conversion to nonproducers. Dimensionless mutation rate, A , is seen to have a tremendous
impact on the "lag phase" before an exponential conversion to nonproducers occurs.
1
0.8
to use high-serum-containing medium for preparation of an inoculum before switching to low-serum or serum-free medium during the production stage. If a condition favoring P over N P cannot be found, one must cultivate cells in a batch mode before the fraction of N P becomes high. When A S L 1,F increases very slowly in the beginning and increases very rapidly thereafter (Figure 2). Therefore, MAb production should be completed in limited generations depending on the acceptable value range of F. For the cultivation of cells in a 10 000-L bioreactor, 40-50 generations from a single cell are needed. When A = and S = 1.2, F will be over 0.9 after 50 generations. Therefore, the decision of production scale should be made considering A and S. Alternatively, the production scale of 10 000 L can be accepted by using a pure inoculum. To produce 2 X 106 cells from a single cell requires about 21 generations, while another 23 generations are needed from 1 mL of 2 X lo6 cells/mL to 10 000 L of 2 X lo6 cells/mL. Therefore, if the culture is started with 2 X lo6 cells of 100% P, MAb production on a 10 000-L scale can be made with less than 10%)loss of MAb. However, if the inoculum of 2 X lo6 cells contains 1% N P , MAb production on a 10 000-L scale results in about 20% loss of MAb. This theoretical result demonstrates that MAb production can be very sensitive to the purity of the inoculum. Pure P may be prepared by sorting P from the mixed population of P and N P by using flow cytometry with proper immunofluorescent staining (Ozturk and Palsson, 1990a; Shapiro, 1988). Although long-term cultivation using free cell suspension culture with stable MAb production cannot be achieved under the condition of A S 2 1, it is possible with immobilized cells. Immobilized cells in low-protein medium can divide, but their growth rate is almost zero when
+
I
0.0 0
100
50
150
n Figure 3. Fraction of nonproducer during long-term cultivation of hybridoma cells. Open circle represents experimental data of
Ozturk and Palsson (1990b). Solid line represents the simulated result with A = 0.0017 and 5' = 1.072. A and S were obtained by using MINPACK with two-parameter estimation. important to find culture conditions that are more favorable for the growth of P over NP. The nonproducer does not necessarily have a growth advantage over P due to a reduced metabolic load, since according to theoretical calculations metabolic load for MAb synthesis has small effects on the growth rate (Savinell and Palsson, 1990a,b). Experience with two murine cell lines shows that highserum-containing medium provides a more favorable growth condition of P as compared to N P , while low-serum or serum-free medium provides a more favorable growth condition for N P than P (Lee and Palsson, 1990; Ozturk and Palsson, 1990b). For such cell lines it is preferable
+
Biotechnol. Prog., 1991, Vol. 7, No. 1
the cell density in the gel beads is maximal (Lee and Palsson, 1990). For example, with a doubling time of 24 h, cells in free suspension culture experience 30 generations within a 1-month cultivation. However, if the immobilized cells have a growth rate reduced 10-fold, they experience only three generations within the same 1-month cultivation. T h e reduced number of generations will result in a lower fraction of NP in the total population. In addition, concentrations of any autocrine growth factors inside the gel beads will be high due to the high local cell density (on the order of lo7 cells/mL), and these factors may reduce the mutation rate, thus reducing A. One such autocrine growth factor is interleukin-6 (IL-6), known t o stimulate immunoglobulin synthesis in lymphoid cells (Aarden, 1989; Hirano e t al., 1986). In conclusion, the simple dynamic model presented illustrates t h e sensitivity of the balance between producing and nonproducing hybridoma cells in long-term cultures t o their relative growth rates at low mutation rates. The analysis helps in guiding the development of production strategies for monoclonal antibodies.
Notation cultivation time, h cell density of P, cells/mL cell density of NP, cells/mL initial cell density of P, cells/mL initial cell density of NP, cells/mL rate of loss of antibody productivity, h-1 specific growth rate of P , h-1 specific growth rate of NP, h-l probability of loss of MAb productivity dimensionless rate of loss of MAb productivity (a/ PP)
F
fraction of N P in a total population [Xn/(Xp+
n XP Xn S
generation number (r/ln 2) dimensionless cell density of producer (xp/xpo) dimensionless cell density of nonproducer (xn/xpo) dimensionless specific growth rate (~,,/w~) dimensionless cultivation time (ppt)
X,) 1
T
Acknowledgment This research was supported by the National Science Foundation EET-8712765). We thank Alice Chuck and Joanne Savinell for their valuable discussions.
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Garbow, B. S.;Hillstrom, K. E.; More, J. J. MINPACK subroutines; Argonne National Laboratory: Argonne, IL, 1980. Cardner, J. S.; Chiu, A. L. H.; Maki, N. E.; Harris, J. F. A Quantitative Stability Analysis of Human Monoclonal Antibody Production by Heteromyeloma Hybridomas, Using an Immunofluorescent Technique. J. Immunol. Methods 1985, 85, 335-346. Goding, J. W. Antibody Production by Hybridomas. J. Immunol. Methods 1980, 39, 285-308. Heath, C.; Dilwith, R.; Belfort, G. Methods for Increasing Monoclonal Antibody Production in Suspension and Entrapped Cell Cultures: Biochemical and Flowcytometric Analysis as a Function of Medium Serum Content. J . Biotechnol. 1990,15, 71-90. Hengartner, H.; Meo, T.; Muller, E. Assignment of Genes for Immunoglobulin K and Heavy Chains to Chromosomes 6 and 12 in Mouse. Proc. Natl. Acad. Sci. U.S.A. 1978,4494-4498. Hirano, T.; Yasukawa, K.; Harada, H.; Taga, T.; Watanabe, Y.; Matsuda, T.; Kashiwamura, S.; Nakajima, K.; Koyama, K.; Iwamatsu, A.; Tsunasawa, S.; Sakiyama, F.; Matsui, H.; Takahara, Y .;Taniguchi,T.; Kishimoto, T. Complementary DNA for a Novel Human Interleukin (BSF-2) that induces B Lymphocytes to Produce Immunoglobulin. Nature 1986,324, 73-76. Imanaka, T.; Aiba, S. A Perspective on the Application of Genetic Engineering: Stability of Recombinant Plasmid. Ann. N . Y . Acad. Sci. 1981, 369, 1-14. Lee, G. M.; Palsson, B. 0. Immobilization Can Improve the Stability of Hybridoma Antibody Productivity in Serum-free Media. Biotechnol. Bioeng. 1990, 36, 1049-1055. Lemke, H.; Hammerling, G. J.; Hammerling, U. Fine Specificity Analysis with Monoclonal Antibodies of Antigens Controlled by the Major Histocompatibility Complex and by the Qa/TL Region in Mice. Immunol. Rev. 1979, 47, 175-206. Moser, H. Structure and Dynamics of Bacterial Populations Maintained in the Chemostat. Cold Spring Harbor Symp. Quant. Biol. 1957, 22, 121-137. Ozturk, S. S.; Palsson, B. 0. Isolation and Characterization of Monoclonal Antibody Producing and Nonproducing Cell Populations. Presented at the 200th National Meeting of the American Chemical Society, Washington, DC, August 1990a. Ozturk, S. S.;Palsson, B. 0.Loss of Antibody Productivity During Long-term Cultivation of Hybridoma Cell Line in Low Serum and Serum-free Media. Hybridoma 1990b, 9, 167-175. Rubinow, S. I. In Introduction to Mathematical Biology; John Wiley & Sons: New York, 1975; pp 31-37. Savinell, J. M.; Palsson, B. 0. Estimation of Metabolic Resource Distributions Using Linear Optimization Theory. Presented at the Annual AIChE Meeting, Chicago, IL, November 1990a. Savinell, J. M.; Palsson, B. 0. Network Analysis of Intermediary Metabolism Using Linear Optimization: 11. Interpretation of Hybridoma Cell Metabolism. J . Theor. Biol. 1990b, submitted for publication. Shapiro, H. M. In Practical Flow Cytometry;Alan R. Liss, Inc.: New York, 1988; pp 282-283. Westerwoudt, R. J.;Naipal, A. M.; Harrisson, C. M. H. Improved Fusion Technique. 11. Stability and Purity of Hybrid Clones. J . Immunol. Methods 1984, 68, 89-101. Wilde, C. D.; Milstein, C. Analysis of Immunoglobulin Chain Secretion Using Hybrid Myelomas. Eur. J . Immunol. 1980, 10,462-467. Accepted November 9, 1990.
