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JOURNALOF FERMENTATION ND BIOENGINEERINGVol. 80, No. 5, 520-521. 1995

TECHNICAL NOTEEstimating Segregational Plasmid Instability in Recombinant Cell

Cultures: A Generalized ApproachPINAKI BHATTACHARYA* AND DEBASHIS ROY

Chemical Engineering Department, Jadavpur University, Calcutta 700 032, IndiaReceived 1 May 1995/Accepted 5 September 1995

A generalized methodology for estimation of the relative segregation rate, @), in a recombinant cell culturehas been presented that does not require the a priori assumption of a constant p. Moreover, measurement ofno other variable in addition to those conventionally monitored, viz. overall culture concentration and fractionplasmid-bearing cells is necessary.

[Key words: recombinant cell, plasmid instability]

The fundamental rate equations describing the growthkinetics 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 denoterespectively specific growth rate and cell concentration,and superscripts + and - refer to the plasmid-bearingand plasmidless cells respectively.Analytical integration of Eqs. 1 and 2 yielding solu-tions for X+(t) and X-(t) is possible only when boththe following conditions are satisfied: (a) cell growth isexponential, i.e. p+ and pP are constant; (b) parameterp is time-invariant and independent of /-1. Often condi-tion (b) is assumed a priori and the integrated form ofEq. 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 toestimate p from the slope of the straight line obtainedby plotting In [X+/X&] versus t in accordance with Eq.la. The assumption of a time-invariant p is obviouslyimplicit in this approach.An alternative procedure is outlined below that isbased on the suggestion of DiBiasio and Sardonini (2) inwhich segregational plasmid instability may be assumedto be dependent on growth rate. It will be seen laterthat this premise does not result in loss of generality ofthe proposed methodology since it is also applicable tothose systems where plasmid instability is independent ofgrowth 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 Monodfunctions of a single limiting feed substrate, S, (say),then,U+=,&[s/(s+KJ] (7a)

andP ~ =/-l;[S/(S+&)] Ub)

with the saturation constant K, usually having the samevalue for both recombinant and segregant species. Theratio of the specific growth rates is thus equal to theratio of the maximum specific growth rates-a constantfor the system, i.e.

,Lf /,Lf+=#L&/&=aBy 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 possibleto estimate p as a function of p+ corresponding to theinstants at which x and F are measured experimentallyfrom O.D. measurements and agar plating, respectively.It should be noted here that Eq. 8, which expressesthe fact that cy is constant, depends on the assumptionthat both the recombinant and segregant species haveidentical values of KS. However, in some recombinantcells the uptake and assimilation of growth-limiting sub-strate are dependent on the presence of a specific geneproduct encoded on the plasmid gene, e.g. toluene con-sumption (and thus KS for toluene) depends upon thepresence of Tol-plasmid in Pseudomonas. The value ofKS for such recombinant cells is, therefore, differentfrom 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

woo0 . G 2 . 0 4 . 0 6 . 0 8 . 0 1 0 . 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 thoseconditions only in which growth-limiting substrate is inexcess (i.e. S>>ZQ, so that inspite of differing K, valuesfor the recombinant and segregant strains, Eq. 8 maystill 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 experimentwith segregant cells only (It is worthwhile to note herethat even if p is assumed constant, it is generally notpossible to estimate /*+, p- and p from the same experi-ment). (ii) In the actual experiment with recombinantcells, measure the value of F at the start of exponentialgrowth (t=to, say) in batch culture (i.e. D=O) by count-ing number of colonies on agar plates with and withouta marker reagent, and let F. denote the value of F. Atthis 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 calculateddirectly. 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 beenmeasured 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 maybe prepared depending on whether the objective is tocorrelate p and p+ or simply to observe the variationaltrends in the progressive values of p in the course ofthe fermentation.Even though the method described above appears tobe simple and straightforward, it is potentially usefulfrom the reaction engineering viewpoint since it providesa realistic estimate of the crucial plasmid instability

parameter that determines system productivity.Moreover, the proposed methodology also provides anindirect method for evaluation of the specific growthrates, pLf and 1_1--, eyond the exponential growth phase,Le. when their values are changing with time, withoutnecessitating measurement of the limiting substrateconcentration 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 beexpected to remain constant as well. However, for agiven host-vector system, the value of pm+ is a functionof culture conditions, e.g. temperature, pH etc. Thecorresponding value of p should therefore be a functionof culture conditions too. Thus the approach describedherein can be utilised to obtain quantitative informationon the dependence of p on culture conditions whichshould be of immense importance in subsequent engineer-ing design work.However, the absence of any valid correlation betweenp and p+ cannot be ruled out as an a priori. Theremay indeed be systems, as indicated earlier, where thenature of variation of p may be perfectly random, evenin the exponential growth phase. Subsequently, depend-ing on the nature of the trend observed in progressivevalues of p in the course of a recombinant cell fermen-tation, host-vector systems can broadly be categorisedinto two types, viz. Type l-where a definite correlationexists between p and p+ and Type 2-where p cannotbe correlated with p+, the nature of its variation beingapparently random. To ascertain which category a par-ticular system belongs to, experiments must be per-formed and the data analysed in accordance with thescheme outlined in this report. In Fig. 1 is plotted a typi-cal set of p vs. ,u+ data for both types of systems; forthe Type 1 case it is assumed that {(l-p)/p+} is con-stant whereas for Type 2 the p values plotted representa population of random numbers with a mean of 0.15(the same as the Type 1 value of p corresponding toconstant p+) and maximum variation of *50x aboutthe mean.Recombinant cell systems may also be characterizedon the basis of the effect of dilution rate variations (incontinuous culture) on the value of p. Since productiv-ity of a bioreactor harbouring a recombinant cell culturedepends substantially on the magnitude and variationaltrends of the relative segregation rate, the importance ofsuch characterizations cannot be overestimated from thereaction engineering perspective.

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

1. Imanaka, T. and Aiba, S.: A perspective on the application ofgenetic engineering: stability of recombinant plasmid. Ann.N.Y. Acad. Sci., 369, 1-14 (1981).2. DiBiasio, D. and Sardonini, C.: Stability of continuous culturewith recombinant organisms. Ann. N.Y. Acad. Sci., 469, lll-117 (1986).