JOURNAL OF FERMENTATION AND BIOENGINEERING Vol. 80, No. 5, 520-521. 1995

TECHNICAL NOTE

Estimating Segregational Plasmid Instability in Recombinant Cell Cultures: A Generalized Approach

PINAKI BHATTACHARYA* AND DEBASHIS ROY

Chemical Engineering Department, Jadavpur University, Calcutta 700 032, India

Received 1 May 1995/Accepted 5 September 1995

A generalized methodology for estimation of the relative segregation rate, @), in a recombinant cell culture has been presented that does not require the a priori assumption of a constant ‘p’. Moreover, measurement of no other variable in addition to those conventionally monitored, viz. overall culture concentration and fraction plasmid-bearing cells is necessary.

[Key words: recombinant cell, plasmid instability]

The fundamental rate equations describing the growth kinetics of a recombinant cell culture (1) are:

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

(dAp/dt)=p,utX+ +p-X- -DX- (2)

where ‘p’ is the relative segregation rate, D is the dilu- tion rate (=0 for a batch culture), p and X denote respectively specific growth rate and cell concentration, and superscripts + and - refer to the plasmid-bearing and plasmidless cells respectively.

Analytical integration of Eqs. 1 and 2 yielding solu- tions for X+(t) and X-(t) is possible only when both the following conditions are satisfied: (a) cell growth is exponential, i.e. p+ and pP are constant; (b) parameter ‘p’ is time-invariant and independent of /-1. Often condi- tion (b) is assumed a priori and the integrated form of Eq. 1, i.e.

In [X+/X&1=(1-p)p+t (la)

(where XJ is the value of X+ at time t=O), is used to estimate ‘p’ from the slope of the straight line obtained by plotting In [X+/X&] versus t in accordance with Eq. la. The assumption of a time-invariant ‘p’ is obviously implicit in this approach.

An alternative procedure is outlined below that is based on the suggestion of DiBiasio and Sardonini (2) in which segregational plasmid instability may be assumed to be dependent on growth rate. It will be seen later that this premise does not result in loss of generality of the proposed methodology since it is also applicable to those systems where plasmid instability is independent of growth rate.

Overall culture concentration Xr may be defined as

Xr=x++x (3)

Differentiation with respect to time gives

(dX/dt) = (dX+/dt) + (dZ/dt) (4)

Substitution from (1) and (2) in (4) gives

(w/dt) = 11 +X+ + ,u -XP ~ DXr (5)

* Corresponding author.

On dividing by x and substituting F=X+/S, (i.e. F=fraction of plasmid-bearing cells), Eq. 5 becomes

(l/Xr)(dXr/dt)+D=F/c++(l-F)p (6)

Now, if the specific growth rates p+, /1 are Monod functions of a single limiting feed substrate, S, (say), then

,U+ =,&[s/(s+KJ] (7a)

and

P ~ =/-l;[S/(S+&)] U’b)

with the saturation constant K, usually having the same value for both recombinant and segregant species. The ratio of the specific growth rates is thus equal to the ratio of the maximum specific growth rates-a constant for the system, i.e.

,Lf /,Lf’+=#L&/&=a

By defining the function g[x(t)] as

g[x(t)] =(l/x)(dx/dt)

Equations 6 and 1 can now be rewritten as

g(Xr)+D=$[F+(l-flu]

and

(8)

(9)

(10)

g(X+)+D=pt(l-p) (11)

With the help of these Eqs., i.e. 10 and 11, it is possible to estimate ‘p’ as a function of p+ corresponding to the instants at which x and F are measured experimentally from O.D. measurements and agar plating, respectively.

It should be noted here that Eq. 8, which expresses the fact that cy is constant, depends on the assumption that both the recombinant and segregant species have identical values of KS. However, in some recombinant cells the uptake and assimilation of growth-limiting sub- strate are dependent on the presence of a specific gene product encoded on the plasmid gene, e.g. toluene con- sumption (and thus KS for toluene) depends upon the presence of Tol-plasmid in Pseudomonas. The value of KS for such recombinant cells is, therefore, different from that for the corresponding segregant cells. Conse- quently, for such host-vector systems the use of Eqs. 10

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VOL. 80, 1995 TECHNICAL NOTES 521

woo 0.G 2.0 4.0 6.0 8.0 10.0

Time (h)

FIG. 1. Illustration of Type 1 and Type 2 systems. 0, Sp. gr. rate (h-l); x , p (Type 1); 0 , p (Type 2).

and 11 to estimate ‘p’ and /*+ values is limited for those conditions only in which growth-limiting substrate is in excess (i.e. S>>ZQ, so that inspite of differing K, values for the recombinant and segregant strains, Eq. 8 may still be obtained from Eqs. 7a and 7b.

A scheme for evaluation of the parameter (Y is out- lined below: (i) Estimate p; from a separate experiment with segregant cells only (It is worthwhile to note here that even if ‘p’ is assumed constant, it is generally not possible to estimate /*+, p- and p from the same experi- ment). (ii) In the actual experiment with recombinant cells, measure the value of F at the start of exponential growth (t=to, say) in batch culture (i.e. D=O) by count- ing number of colonies on agar plates with and without a ‘marker’ reagent, and let F. denote the value of F. At this instant, p + =& and P- =I*;. Consequently, at t= to, Eq. 10 reduces to the form

s(~T)t=to=~,+Fo+~,(l --Fo) (lOa)

In Eq. 10a the only unknown is ,u,$ which is calculated directly. Thus cy is obtained.

Now, once (Y is known, the value of p+ can be ob- tained at all the instants at which F and P have been measured using Eq. 10. Again X+ =FS. Therefore (l- p)p+ can also be subsequently estimated using Eq. 11. Finally, a plot of (1 -p),u+ versus p+, or ‘p’ versus t may be prepared depending on whether the objective is to correlate ‘p’ and p”+ or simply to observe the variational trends in the progressive values of ‘p’ in the course of the fermentation.

Even though the method described above appears to be simple and straightforward, it is potentially useful from the reaction engineering viewpoint since it provides a realistic estimate of the crucial plasmid instability

parameter that determines system productivity. Moreover, the proposed methodology also provides an indirect method for evaluation of the specific growth rates, pLf and 1_1--, beyond the exponential growth phase, Le. when their values are changing with time, without necessitating measurement of the limiting substrate concentration as required by the Monod Eqs. 7a and 7b.

If ‘p’ is indeed a function of p+, then in the exponen- tial growth phase when ,u + =p,$ a constant, ‘p’ may be expected to remain constant as well. However, for a given host-vector system, the value of pm+ is a function of culture conditions, e.g. temperature, pH etc. The corresponding value of ‘p’ should therefore be a function of culture conditions too. Thus the approach described herein can be utilised to obtain quantitative information on the dependence of ‘p’ on culture conditions which should be of immense importance in subsequent engineer- ing design work.

However, the absence of any valid correlation between ‘p’ and p+ cannot be ruled out as an a priori. There may indeed be systems, as indicated earlier, where the nature of variation of ‘p’ may be perfectly random, even in the exponential growth phase. Subsequently, depend- ing on the nature of the trend observed in progressive values of ‘p’ in the course of a recombinant cell fermen- tation, host-vector systems can broadly be categorised into two types, viz. Type l-where a definite correlation exists between ‘p’ and p’+ and Type 2-where ‘p’ cannot be correlated with p+, the nature of its variation being apparently random. To ascertain which category a par- ticular system belongs to, experiments must be per- formed and the data analysed in accordance with the scheme outlined in this report. In Fig. 1 is plotted a typi- cal set of ‘p’ vs. ,u+ data for both types of systems; for the Type 1 case it is assumed that {(l-p)/p+} is con- stant whereas for Type 2 the ‘p’ values plotted represent a population of random numbers with a mean of 0.15 (the same as the Type 1 value of ‘p’ corresponding to constant p+) and maximum variation of *50x about the mean.

Recombinant cell systems may also be characterized on the basis of the effect of dilution rate variations (in continuous culture) on the value of ‘p’. Since productiv- ity of a bioreactor harbouring a recombinant cell culture depends substantially on the magnitude and variational trends of the relative segregation rate, the importance of such characterizations cannot be overestimated from the reaction engineering perspective.

One of the authors (D. Roy) is grateful to Jadavpur University for the grant of a research fellowship.

REFERENCES

1. Imanaka, T. and Aiba, S.: A perspective on the application of genetic engineering: stability of recombinant plasmid. Ann. N.Y. Acad. Sci., 369, 1-14 (1981).

2. DiBiasio, D. and Sardonini, C.: Stability of continuous culture with recombinant organisms. Ann. N.Y. Acad. Sci., 469, lll- 117 (1986).