ELSEVIER Biochemical Engineering Journal 1 ( 1998) 219-223

Biochemical Engineering Journal

A method for estimation of growth and instability parameters of batch recombinant cultures

Debashis Roy, Pinaki Bhattacharya * Deparrment of Chemical Engineering, Jadavpur University Calcutta, Calcutta 700 032, India

Received 20 December 1997; accepted 4 March 1998

Abstract

We describe a method for estimation of kinetic parameters quantifying cell growth and segregational instability during exponential growth of batch recombinant cultures-when these parameters may be considered constant, thereby permitting analytical solution of the kinetic model equations. It is shown that suitable regrouping of the original unknown parameters along with a judicious restatement of the original nonlinear objective function allows the formulation of a one-parameter search algorithm based on linear least-squares analysis, not necessitating linearization of the model using approximate methods. Further, confinement of the searched-for parameter within a known, bounded interval guarantees convergence of the search. Using published experimental data, it is demonstrated that a user-interactive computer program titled RECOMB, based on the proposed algorithm, may be conveniently employed to obtain the required parameter estimates using data from a single mixed-culture experiment. 0 1998 Elsevier Science S.A. All rights reserved.

Keywords: Instability parameters; Cell growth; Batch recombinant cultures

1. Introduction

For kinetic analysis of a recombinant cell systemconsisting of plasmid-bearing (P’) and plasmidless (P-) cells from a distributed approach, the model equations due to Imanaka and Aiba [ 1 ] may be considered:

dX+/dt=( I-p)p+X+ -DX+ (1)

dX-ldt=pp+X+ +/CX- - DX- (2)

In the above equations, ‘x’ denotes cell concentration (in appropriate units), p denotes specific growth rate [time- ‘1, D denotes dilution rate [time- ’ ] (for batch cultures D = 0)) superscripts + and - refer to P+ and P- species, respec- tively, and the parameter p is the relative segregation rate, which quantifies the segregational plasmid instability. Given a set of experimentally determined values of X+ and X- against time t, the problem is to evaluate the unknown kinetic parameters p, $ and p-, using the above model equations, for batch cultures.

The objective of a particular parameter estimation study dictates the appropriate experimental conditions, for, it is the latter that determine whether, if at all, any or all of the unknown parameters may be considered constant. For exam-

* Corresponding author.

1369-703X/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PlIS1369-703X(98)00010-2

ple, during the development of a new host-vector system intended to be employed for a specific target job, it is neces- sary to obtain a preliminary quantitative idea of its growth and stability characteristics for different possible combina- tions of cultivation conditions and/or culture media. In such a situation, it would be advantageous to have a parameter estimation protocol that requires minimum experimentation and yet yields the desired parameter estimates with speed and accuracy. Arguably, the most convenient of such protocols would involve a simple batch fermentation experiment (e.g., in a shake flask) with a mixed culture of the Pf and P- strains, for each possible set of conditions, and a ready-to- use computer program to subsequently process the experi- mental data and directly provide the desired estimates. As will be evident from the discussions that follow, the meth- odology developed in this study involves a single mixed- culture batch experiment, and direct input of this data to a computer program employing the proposed technique pro- vides almost immediately the required parameter estimates.

A pertinent question which arises is regarding the con- stancy/variability of the unknown parameters in the model equations. From the viewpoint of mathematical rigour, an analytical solution (assuming it is possible) to the model equations-considered as the ‘objective function’ for subse- quent regression analysis, is definitely preferable to an

220 D. Roy, P. Bhattacharya /Biochemical Engineering Journal 1 (1998) 219-223

approximate solution. By inspection of the model equations (i.e., Eqs. ( 1) and (2) ) it is evident that an analytical solution is impossible unless the unknown parameters are all assumed constant. During exponential growth in batch culture, it is reasonable to consider the growth parameters p+ and p- constant. Now, the segregational instability parameter, p, is defined as the ratio of the rate of formation of P- cells to the rate of growth of P+ cells. Although this definition does not strictly imply that p should be a function of p+, it is highly probable that such a functionality exists-in fact, results from earlier experimental studies [ 2,3] lend credence to such an inference. Consequently, for exponential growth in batch cul- ture where p + is constant, it does not seem unrealistic to consider p simultaneously invariant as well. Accordingly, for the present analysis, the unknown parameters p, p+ and p- are regarded as constants.

It is worthwhile to mention here that methodologies have been reported earlier ( [ 4,5] ) for estimating the parameters p, IL+ and ru-, using Eqs. ( 1) and (2)) considering them as variables. However, due to the lack of an analytical solution to the model equations, these methods must rely on approx- imate techniques, e.g., numerical/graphical differentiation, that makes their reliability suspect. In addition, the unavoid- able computational burden associated with approximate solutions becomes a cause for inconvenience as well. Accordingly, for the case considered in this study, it is both advantageous as well as justifiable to consider the parameters p, pf and p- constant.

2. Analysis

For constant p, pFL+ and p- the analytical solution to Eqs. ( 1) and (2) may be expressed as:

ln[(llF)-l+A]=ln[A+(XJXi)]+(p- -p++pp+)t

(3)

where F denotes fraction P+ cells, i.e., F=X+/XT, (XT= X++X-);parameterAisdefinedasA=pp+I(p--p++ pp+); and the subscript ‘0’ denotes initial conditions. The original model, consisting of two simultaneous ODES is thus transformed to a single algebraic equation (i.e., Eq. (3) >. It is known that using methods of regression analysis, it is possible to estimate the values of ‘k’ number of unknown parameters (in Eq. (3)) k= 3) from a given model equation (linear or nonlinear) as long as a minimum of (k + 1) exper- imental data points are available.

Using the substitutions f= X- /X+ = ( 1 lF) - 1 and B = ( pL- - p+ + pp+ ) Eq. (3) is further simplified to the form

ln(A+f)=ln(A+f,)+Bt (4)

The problem now is to estimate the constants A and B (which contain the unknown kinetic parameters) from the givenfvs. t data. It is interesting to note here that although the original

objective function contains three unknown parameters (p+, pL- and p) , regrouping them in the form ofA and Beffectively transforms the original three-parameter problem to an equiv- alent two-parameter form, which is definitely advantageous from the viewpoint of regression analysis. Further, parameter A is so defined that its values are confined within a specified, bounded interval: since p- > pf in general, and 0 I p I 1 by definition, it follows that 0 IA I 1. Now, the classical linear least-squares method is clearly inapplicable for this problem since the constants A and B do not appear linearly either in the original model equation, (i.e., Eq. (4)) or in any trans- formed form of the same [ 61. The situation seemingly calls for an approach involving nonlinear least-squares, using some method of nonlinear regression analysis, e.g., the Gauss-Newton method, the method of steepest descent, the Marquardt method, etc.

However, all these methods, which are based on the prin- ciple of minimising the SSR (sum of squared residuals) in order to obtain estimates of the parameter vector, have some intrinsic lacunae. They are, in essence, approximate methods based on linearization of a nonlinear objective function using a truncated Taylor series expansion, and will not work satis- factorily if the degree of nonlinearity in the objective function is so high that Taylor series approximation is inadequate. Further, these methods require an initial guess of the param- eter vector, and unless these guesses are so ‘chosen’ that the initial starting point for the iterative search (towards min- imising the SSR) is ‘sufficiently’ close to the unknown target point, the search will not converge and the method will prove abortive. Given these constraints, it does not seem particu- larly instructive to follow a nonlinear least squares approach if an alternative method may be devised somehow that is free from these drawbacks. As demonstrated below, this is achiev- able, to begin with, by an equivalent restatement of the orig- inal objective function, Eq. (4)) as follows:

Substituting

y=ln(A+f)

Eq. (4) becomes,

(5)

y=yo+Bt (6)

Now, if a trial value, A,,, be assigned to parameter A, then, corresponding to this value, a set of ‘y’ values may be gen- erated using Eq. (5). Plotting these y values against the cor- responding t values gives a straight line in accordance with Eq. (6). The least-squares estimates of the intercept y0 and slope B for the best fitting straight line are:

(7)

(8)

where ‘n’ denotes the number of data points. Once y. is obtained, A may be recalculated from Eq. (5) by substituting y = yof=fo. Let A,,, denote this recalculated value. The close-

D. Roy, P. Bhattacharya / Biochemical Engineering Journal I (1998) 219-223 221

ness of Acal to A,,, determines the degree of convergence of the proposed search algorithm. A convergence parameter, c, is defined as

With respect to a small pre-specified positive fraction 71, say, the convergence criterion is then expressed as I< 7. If this criterion is satisfied, the trial is terminated, else it is to be continued starting with a new value of A,,, until the desired convergence (i.e., 5 77) is achieved.

As explained earlier, possible values of parameter A are confined in the interval [ 11. Thus, it is only necessary to choose a step size ( = SA, say) to conduct the search. At any point during the search, if a change of sign of the quantity (Atr, -Acal) occurs between successive trial values Atr,* and (A,,,*+SA), it indicates that the target value lies between these trial values. The trial is then re-started from A,,,* with a reduced step-size (usually = SA/ 10) in order to locate the point at which the stipulated degree of convergence is attained.

Once A and B have been estimated, it is easy to determine p, CL+ and CL-. The parameters A and B have been so defined that pp+ =AB. Thus, the value of p/.~+ is known. Again, integrating Eq. ( 1) one obtains

(llt)In(X+/X~)=~+ -pp+ (10)

in which pf being the only unknown, is evaluated. From the values of CL+ and pb+, p is determined. Finally, using the defining expression for parameter A, the value of p- is obtained.

The above algorithm has been employed to develop a user- interactive computer program RECOMB. This program, written in the BASIC programming language, may be used to obtain estimates of the kinetic parameters (viz., II+, p- and p) directly from the input experimental data (t, XT, F) after the value of v and the number of trial steps have been specified by the user.

3. Illustrative example

Published data [ 71 from the exponential growth phase of a mixed batch culture of wild-type and recombinant Escher- ichia coli strains, is used to illustrate the usefulness of the proposed method. The said data, used as input to RECOMB, is shown graphically in Fig. 1. Confining parameter A in the interval [ 11, parameter estimates are obtained by considering progressively decreasing values of 77 between 0.1 and 0.0001, for maximum number of steps = 100 as well as = 1000. Table 1 contains these estimates as well as those reported earlier [ 71. The accuracy of parameter estimation is expected to increase with decreasing values of 7. This is evident from Table 1, although use of a value 77 <O.OOOl is seemingly unnecessary for the example considered.

0

0 0

0

o 0

0

0

0 0

O.OO~

timk (h) ’ 0

Fig. 1. Variation of culture concentration, XT (OD 600 nm) and fraction P+ cells, F with time (h) during exponential growth of a mixed culture of wild- type and recombinant E. coli (Bentley and Kampala [ 71). 0000, culture concentration (OD 600 nm): p; q CIUO, fraction P+ cells: F.

Table 1 Comparative values of parameter estimates obtained in the present and earlier a studies

(a) Present study (using RECOMB, with 0 <A < 1)

7 CL+ 0-l) CL- (h-l)

(i) Maximum number of trial steps = 100 0.1 0.4108 0.5052 0.01 0.4105 0.5070 0.001 0.4105 0.5071 0.0001 0.4105 0.507 1

(ii) Maximum number of trial steps = 1000 0.1 0.4105 0.5069 0.01 0.4105 0.5070 0.001 0.4105 0.5071 0.0001 0.4105 0.507 1

(b) Earlier study [Ref. [7]]

P

0.02700 0.02620 0.02615 0.02615

0.02620 0.02620 0.02615 0.026 I5

CL+ (h-l) CL- (h-l)

Values f 95% limits

“Ref. [7].

0.417 0.511 0.006 0.008

4. Conclusions

From the results it is evident that the estimates of p+ and p- obtained from the mixed-culture experiment data using RECOMB are in excellent agreement with those reported earlier [7]. Regarding the value of p, however, it may be recalled that in view of the high copy number of the plasmid used, the earlier authors [ 71 assumed a priori that the value

222 D. Roy, P. Bhattachatya / Biochemical Engineering Journal I (1998) 219-223

of p would not be significantly different from zero. This a priori premise of a negligible value of p led them to the simplifying assumption

pp+I(p- -EL+)“1 (11)

which ultimately serves to eliminate one of the three unknown parameters (i.e., p) from the problem. Using the estimates of p”+, p- and p obtained in the present study, where no such a priori assumption was made regarding the value of p, it is seen that the ratio pp+ /( p- - p+ ) has a value of the order of lo%, which evidently suggests that, in the context of the present example, the condition expressed in Eq. ( 11) above seems to be inappropriate.

Perhaps because the objective of the experimental study [ 71 was primarily instructional, (in fact Eq. (3) was used not as an objective function for parameter estimation but rather for purposes of kinetic characterization of the given host-vector system), two additional experiments were performed for estimating the parameters CL+ and p- inde- pendently of the mixed-culture experiment. The latterparam- eter was estimated from an exponentially growing batch culture containing 100% P- cells, whereas the former was measured from an exponentially growing batch culture of Pf cells maintained under selection pressure. The excellent agreement observed between the values of p+ and p- thus obtained, with those estimated from the mixed culture exper- iment using RECOMB leads to the interesting and probably significant conclusion that as long as exponential growthcon- ditions hold, that is competition for nutritional resources is insignificant, the growth rates of two coexisting strains (here P+ and P-) in a culture are independent of one another.

Thus, for the parameter estimation problem under consid- eration, it is demonstrated that suitable regrouping of the original unknown parameters along with a judicious restate- ment of the original nonlinear objective function allows the formulation of a one-parameter search algorithm, (where no guess of the other parameter, i.e., B is required) based on linear least-squares analysis, in which convergence is guar- anteed as the searched-for parameter (i.e., A) is constrained in a known, bounded interval. It is also evident from the illustrative example considered above, that, the computerpro- gram RECOMB-based on this algorithm, is a convenient and reliable tool for directly estimating the kinetic parameters of interest from experimental data.

5. Nomenclature

Parameter defined in terms of the unknown kinetic parametersp,pFL+ andp- (=pp+l(p--p++pp+)) Trial value of the parameter A Recalculated value of the parameter A following the proposed protocol Trial value immediately after which the difference (A,,, -Acal) changes sign Parameter defined such that AB = pp+

f” fo F

;+ P- t X

X+ X- Y Yo

Dilution rate (h- ’ ) Ratio of concentrations of P- and P+ cells (=X-/X+) Initial value of parameter f (i.e., at t = 0) Fraction P+ cells ( =Xf /XT) Number of experimental data points Plasmid-bearing cells Plasmidless cells Real time (h) Concentration of cells in culture (in appropriate units) Concentration of plasmid-bearing cells Concentration of plasmidless cells Parameter defined by Eq. (5) Initial value of parameter y (i.e., at t = 0)

Greek symbols

6A Step-size of A used during trial 77 Small positive fraction used to express the

convergence criterion

P Relative segregation rate of plasmid-bearing cells

P Specific growth rate of cells (h- ’ )

PL+ Specific growth rate of plasmid-bearing cells (h- ’ ) -

F Specific growth rate of plasmidless cells (h- ’ ) Convergence parameter defined by Eq. (9)

Superscripts

+ Plasmid-bearing cells - Plasmidless cells * A particular value T Total, i.e., referring to the whole culture

Subscripts

cal Recalculated value trl Trial value 0 Initial instant (i.e., t = 0)

Acknowledgements

One of the authors (D.R.) is grateful to Jadavpur Univer- sity for grant of a research fellowship.

References

111

121

[31

[41

T. Imanaka, S. Aiba, A perspective on the application of genetic engineering: stability of recombinant plasmid, Ann. New York Acad. Sci. 369 (1981) 1. C.A. Sardonini, D. DiBiasio, A model for growth of Saccharomyces cerevisiae containing a recombinant plasmid in selective media, Bio- technol. Bioeng. 29 (1987) 469. W. Ryan, S.J. Parulekar, Recombinant protein synthesis and plasmid instability in continuous cultures oflkherichia coli JM103 harboring a high copy number plasmid, Biotechnol. Bioeng. 37 (1991) 415. R. Mosrati, N. Nancib, J. Boudrant, Variation and modelling of the probability of plasmid loss as a function of growth rate of plasmid-

D. Roy, P. Bhattacharya /Biochemical Engineering Journal I (1998) 219-223 223

bearing cells of Escherichia coli during continuous cultures, Biotech- [6] J.B. Scarborough, Numerical Mathematical Analysis, 6 edn., Oxford nol. Bioeng. 41 (1993) 395. and JBH Publishing, New Delhi, 1966, pp. 545,556.

[5] P. Bhattacharya, D. Roy, Estimating segregational plasmid instability [7] W.E. Bentley, D.S. Kompala, Plasmid instability in batch cultures of in recombinant cell cultures: a generalized approach, J. Ferment. recombinant bacteria: a laboratory experiment, Chem. Eng. Educ., Bioeng. 80 (5) (1995) 520. Summer 1987, p. 168.