17. Yu. I. Averbukh, N.M, Kostin, and Yu. A. Krylatov, Zh. Prikl. Khim., No. 4, 820-821 (1978).

18. A. A. Baram and O. A. Loshakova, Teor. Osn. Khim. Tekhnol., 12, No. 2, 231-240 (1978). 19. T. M. Ledova, M. A. Balabudkin, and S. A. Plyushkin, Khim. Farm. Zh., No. 5, 96-99

(1980). 20. M. V. Lekveishvili, M. A. Balabudkin, and G. N. Borisov, Soobshch. Akad. Nauk Gruz.

SSR, 81~ No. 3, 621-623 (1976).

USE OF PONTRYAGIN'S MAXIMUM PRINCIPLE FOR SOLVING OPTIMUM CONTROL

PROBLEMS OF THE PROCESSES OF BIOSYNTHESIZING SECONDARY METABOLITES

L. E. Shnaider, V. V. Birykov, and V. M. Kantere

UDC 615.33.012.6

Introduction and Statement of the Problem

To solve the problem of intensifying batch processes of the biosynthesis of secondary metabolites, and especially antibiotics, experimenters frequently come up against the fact that the conditions of fermentation (concentration of nutrients, temperature, pH, etc.) that are optimum for the growth of the producing microorganisms and for the biosynthesis of the desired product differ substantially. Thus, according to Orlova [i], the most intensive growth of the producing agent of oxytetracycline is observed with 200 ~g/ml of organic phos- phorus in the medium, while the rate of synthesis of the antibiotic under these conditions is somewhat less than when the medium contains 50 ~g/ml of phosphorus. In the biosynthesis of penicillin, an excess of glucose, while stimulating the growth of the culture, almost com- pletely suppresses the synthesis of the product [2]. There is similar information in rela- tion to other producing cultures. In view of this, in the optimization of such processes the problem arises of determining the optimum profile of the change in the technological param- eters ensuring an extremum of some criterion of efficiency with the observation of certain limitations. In practice, such problems are usually solved by the empirical method: A nu- trient medium is selected in which the maximum yield of product at the end of fermentation is observed, the concentration of nutrients changing during the process only through their consumption by the culture. No addition of substrate during fermentation is made. However, at the present time in view of the development of regulated fermentation processes the pos- sibility is appearing of adding nutrients during the process and thus controlling it. In general, the statement of the problem of the optimum control of a batch process of fermenta- tion with supplementary additions has been considered by Fishman [3]. Here we shall consider such a problem as applied to actual, fairly simple, structures of a mathematical model.

Let the process of biosynthesizing some product be described by the system of equations

dX dt = ~ (s) x , ( 1 )

clS 1 dt = "-Y P~ ($) X -4- v (t) , (2)

dA at - q (s) x , ( 3 )

where X, A, and S are the concentrations of biomass, of the desired product, and of the limit- ing substrate; ~(S) and q(S) are the specific rates of the growth of the biomass and of the biosynthesis of the products, respectively; Y is the economic factor; and v(t) is the rate of addition of the substrate.

All-Union Scientific-Research Institute of Antibiotics, Moscow. Translated from Khimiko- Farmatsevticheskii Zhurnal, Vol. 15, No. i, pp. 109-113, January, 1980. Original article sub- mitted December 24, 1980.

56 0091-150X/81/1501-0056507.50 �9 1981 Plenum Publishing Corporation

I [

, t q ( 8 ) [

8z #t 8maz ~

Fig. i. Dependence of the specific rates of growth (~) and of the biosynthesis of the products (q) on the concentration of the limiting substrate (S).

Then the optimum control problem can be posed in the following way: to find the initial concentration of substrate So and the program for its addition to the apparatus v(t) ensuring the maximum accumulation of desired product A at a given time T.

Formally

l = A (T) ~ max So, v ( t ) ,

0 < v (0 ~--< Vmax, ( 4 )

x (o) = xo . A (0) = o, S (0) = So,

where T is the given fermentation time; Vma x is the maximum achievable rate of addition~ and I is a criterion of optimality. The dynamics of the process are described by Eqs. (1-3) o

A similar problem was considered by Fishman [3]. Her solution even for the simplest functions ~(S) and q(S) is extremely laborious, requiring the solution of a two-point bound- ary value problem with conditions at the left and right ends of a trajectory [4] and contain- ing "special control" sections. Nevertheless, the problem can be substantially simplified if we take as the controlling action not the rate of addition v(t) but the concentration of limiting substrate, S, directly. In this case, Eq. (2) proves to be superfluous, and the statement of the problem appears as follows:

T

I = A (T) = .[. q (S) Xdt > m a x , o s(t)

dX at = ~ (S) X, (5)

dA dt = q ~ ) x ,

Smtn<S(0<Smax, X(~ =Xo, A~)=0.

On passing from problem (1)-(4) to (5) we have, in essence, introduced the additional assumption that the concentration of the substrate can change as rapidly as we desire, while in the preceding formulation it was considered that a change in S in the direction of a de- crease could take place only through the consumption of the substrate by the culture with a final rate equal to (I/Y)~(S)X. Consequently, problem (5) consists in an extension of prob- lem (1)-(4). In this case, if the solution achieved proves to be physically realizabl 9, it can be stated that the conditions found are the optimum for problem (1)-(4), also. Otherwise, we obtain a solution giving an upper estimate of the criterion of optimality in problem (4). In practice, this solution can be realized approximately, which leads to some worsening of the criterion in comparison with its calculated value.

Problem (5) is a variational problem of optimum control with connections in the form of differential equations. For its solution it is possible to use Pontryagin's maximum princi- p!e [4]. In accordance with the procedure of the maximum principle let us write the Hami!- tonian function for problem (5):

n = r (s) x + , ~ (s) x ( 6 )

The equation of the conjugate variable ~ has the form

d ~ = O H d l - - O X '

whence

with the limiting condition:

57

, ( T ) = O . (8)

The optimum control S* at any moment of time is determined from the condition H(S*) = sup< H, whence, taking into account the fact that X(t) s > O:

S*= a~ maxlq(S)+,(O ~(S)]. (9)

Thus, to determine the optimum program of the change in S(t) it is sufficient, starting from the moment t = T, to integrate Eq. (7) from right to left, since the boundary condition (8) is set at the right-hand side, selecting S in each integration step from condition (9).

For example, let the relations ~(S) and q(S) have the form of piecewise linear functions (Fig. i) described by the equation

"S" fklS, at O < S < S 1 F( ) = t k l S l , at S>Sx,

[k~S, at 0 < 8 ~ 8 t q ( S ) = t k 2 S ~ - - k = ( S - - S 2 ) , at S > S , .

(lo)

(11)

Then from condition (9) we obtain

Sl, at , > kl . . . . S* =

k8 /s~, ~ t , < --E-," (12)

To determine the optimum program of control S*(t) let us make use of the continuous func- tion 9(t) with respect to the time.

Since, from condition (8) 9(T) = 0 < (k~/kl), then, thanks to the continuity of the func- tion 4, the inequality 9(T) < (k3/k:) is valid in any finite time interval t s < t ~< T.

In this interval, from (12), we have S* = $2.

Having integrated differential equation (7) over the interval t s < t ~ T with the bound- ary condition (8), we find the value of t s from the relation } (ts) = (k3/kl), whence:

1 k 3 + k2 ts= T - - ~ l n k2

In the interval 0 ~ t < t s we have ~(t) > (k3/k~), whence S* = S~. Thus, the optimum control program in the given problem has the form (Fig. 2)

S* = IS1, a t 0 ~< t < t5

( S~, at t s <t~<T,

where t s is the moment of "switching" of the equation, which is determined from formula (12).

Allowing for a Limitation on the Concentration of Biomass

In practice, in the solution of such problems it is generally necessary to consider an additional constraint with respect to the mass-exchange possibilities of the apparatus, which, in the simplest case, can be given by the inequality X(t) ~< Xmax, ~ < t < T, or, since, ac- cording to the model adopted, the concentration of biomass does not decrease throughout the process (~(S) ~ 0), we have

X (T) : Xmax. (13)

The sign of strict equality has been given in expression (13), since at X(T) <Xma x this constraint is unimportant and the problem reduces to the preceding one.

Conseraint (13) does not affect the structure of the optimality condition but is expressed only in the boundary condition for the conjugate variable -- Eq. (8) is replaced by the follow- ing: ~(T) = X, where I is a factor determined from condition (13).

Thus, if we add condition (13) to the problem considered above, the optimum control pro- gram will have form (12) although the value of t s in this case is determined from the formula

58

x

I "I

i

/ ' , i I J

1

t s r

Fig. 2

X, 6", A,. nl mg/ml rng/ml

' ? G

~0 ,3,4 _72

i I - 30 16

_ 14

i X z2 ZO o,: zo

8

I0 6 4

f

I02030dOSO607060BOIO0110 I00 150 t, h

Fig. 3

Fig. 2. Structure of the optimum equation.

Fig. 3. Change in the concentration of substrate (S), of biomass (X), and of product (A) under optimum control. Axis of abscissas: time t (in h).

In both the problems considered, control has a "switching" nature, i.e., the concentra- tion of substrate is maintained at the level S~, optimal for the growth of the biomass, for some time, and for the remaining time at level $2, optimal for the biosynthesis of the prod- uct. The numerical values of the parameters of the problem are reflected only in the value of

t s �9

Specification of the Relations ~(S) and q(S) in the Form of "Smooth" Functions

The situation is somewhat more complicated when "smooth" functions are selected to de- scribe the dependences of the specific rates of growth and of the biosynthesis of the product on the concentration of substrate, for example, functions of the form

kl S (s) - k ~ § ( 1 4 )

k= (s) (15) q (S) -- k4 + 8 + k~S~

Nevertheless, in the modeling of microbiological processes it is just such relationships that are usually employed. Let us consider how the optimum control problem is solved in this case.

The statement of the problem is similar to problem (5) with the additional constraint (13), the relations p(S) and q(S) being given by Eqs. (14) and (15). The values of the con- stants in these equations have been taken as k I = 0.15 h -~, k= = 0.5 mg/ml,k3 = 13.1 AU/mg'h , k4 = 0.012 mg/mi, k5 = 4.6 ml/mg, Xo = 2 mg/ml, T = 150 h, Smi n = 0, and Sma x = 1.0 mg/ml.

In this case, the approach to a solution of the problem remains the same as before, but the integration of the differential equation (7) and the maximization of the Hamiltonian (6) must be carried out numerically. The optimum program S*(t) obtained and the corresponding X(t) and A(t) curves are given in Fig. 3. The value of the optimality criterion A(t) = 24~860 AU/ml.

For comparison, let us estimate how much the value of the criterion falls if in the solu- tion of the last problem we use the simplified approach described in our previous paper [5]. The essence of this approach is that the optimum control program is sought in the class of "switching" functions regardless of the form of the relations ~(S) and q(S)~ In these cir- cumstances there is no necessity to use variational methods of optimization, since the prob- lem reduces to determining three scalar magnitudes -- the concentrations of substrate that are

59

the optimum for the growth of the biomass (S~) and the biosynthesis of the product (S2) and the time of "switching" t s. In the example considered, we have S~ = 1.0 mg/ml, $2 = 0.05 mg/ml, t s = 9.5 h, and the value of the criterion A(T) = 22,475 UA/ml. Thus, the use of the simpli- fied approach leads to a 10% decrease in the value of the criterion, which permits this ap- proach to be used in the first stages of work to create a technology for a regulated bio- synthesis, but in the final stage the optimization problem must be solved in the full form.

The authors express their gratitude to A. M. Tsirlin for valuable observations made in a review of the manuscript of the paper.

LITERATURE CITED

i. N. V. Orlova, "An investigation of the physiology of Actinomyces rimosus in connection with the biosynthesis of oxytetracycline and its derivatives," [in Russian], Author's Abstract of Doctoral Dissertation, Moscow (1969).

2. L.M. Lur'e, T. P. Verkhovtseva, and M. M. Levitov, Antibiotiki, No. 4, 291 (1975). 3. V.M. Fishman, "The mathematical description and optimum control of the process of bio-

synthesizing antibiotics," Author's Abstract of Candidate's Dissertation, Moscow (1970). 4. A.M. Tsirlin, V. S. Balakirev, and E. G. Dudnikov, Variational Methods of Optimizing

Controlled Objects [in Russian], Moscow (1976). 5. V.V. Biryukov, L. E. Shnaider, and V. M. Kantere, Khim.-Farm. Zh., 14, No. 6, 82 (1980).

60