Semi optimal feed profiles for inorganic phosphate addition in tetracycline fermentation.
Inorganic phosphate concentration during a tetracycline/puromycin fermentation is a very intriguing factor. Very low dose of phosphate is detrimental to cell growth. Low cell concentration will result in low volumetric productivity in fermentation. On the other hand, phosphate concentration should come down to a very low level before tetracycline production begins. Phosphate is inhibitory to tetracycline formation. Therefore, this pose a nice optimization problem.
Young  observed, in lincomycin fermentation, that phosphate was essential for growth, but it also depressed lincomycin production rates. Similar results have been reported for tetracycline production by a Russian scientist . According to him a low initial rate of the culture growth is an obligatory condition for intensive biosynthesis of tetracycline by Act. aureofaciens. The inorganic phosphorous concentration is a factor limiting the growth rate . According to Weinberg  best product yields are obtained when phosphate concentrations are sub-optimal for vegetative growth. Demain's Group in M.I.T. has observed that candicidin , gramicidin S  and cephalosporin  are subject to phosphate inhibition. Extracellular phosphate remains at very low levels during the entire idiophase of candicidin fermentation. If 10 mM phosphate is added at the start of a candicidin fermentation, extra cellular phosphate is not depleted, growth continues throughout fermentation and no antibiotic synthesis occurs . A similar phenomenon occurs in vancomycin fermentations . Martin  discussed control of antibiotic synthesis by phosphate.
Feed Profile optimization in fed batch reactors
Generally used method for this purpose is the Pontrjagin's maximum principle.
Shnaider  first used this method for solving optimal control problems of the processes of biosynthesis of secondary metabolites.
Modak  used this principle for feed profile optimization for fed batch fermentation. The profiles consisted of a period of maximum feed rate, a period of minimum feed rate and a period of singular feed rate. The optimal control sequences depend on the shapes of dependence of the specific growth and product formation rates ([mu] and [pi]) on the substrate concentration(S). However, Modak had only four material balance equations to consider. It shall be algebraically very difficult to apply same method to a complicated model like ours. In the next paper, Modak  discussed computational algorithms for previously discussed theoretical schemes and gave numerical results for penicillin and bacterial cell mass production. Constantinides [14, 15] used the same principle to obtain optimum temperature profiles for batch penicillin fermentation. Lee and Ramirez  used Pontrjagin's maximum principle to optimize both glucose (food) and inducer (I.P.T.G.) feed rates to fed-batch recombinant fermentations. But calculations become very lengthy and they cover pages after pages in the thesis of Lee, though Lee's model does not differentiate between plasmid bearing (PBC) and plasmid free cells(PFC).
Hilaly  applied the same method for optimization of cellular productivity in industrial micro-algae fermentation. In this case, the specific growth had substrate inhibition as well as inhibition by toxin accumulation. Recently Shin  applied optimal control to a model of recombinant fermentation and shown that when [pi] is linear with respect to [mu] and yield co-efficient is constant, the optimal singular feed rate forces substrate concentration to start near the value at which specific growth rate of PBC cells is maximum and moves in time in the direction of maximizing [[mu].sup.+]/[[mu].sup.-].
However, computational difficulties, algebraic as well as numerical, led-us to search for a method which is simpler and not dependant on the complexity of modelling or on dependence of [mu] and [pi] on substrate concentration. Montague  first applied chemotaxis algorithm for optimization of feed profiles to fed batch bioreactors. Maitra improved upon the chemotaxis algorithm by adding buble-sort to it and applied it for optimization of inducer (I.P.T.G.) feeding to recombinant fermentation  as well as glucose feed addition to penicillin fermentation . In this paper, we shall apply the same method to get semi-optimal feed profile for inorganic phosphate addition to fed-batch tetracycline production. For any theoretical attempt for optimization, a good model of the system is required. There have been some attempts to model tetracycline fermentation.
Votruba  developed a model for tetracycline biosynthesis--this model can be described as follows: for growth and substrate consumptions simple Monod kinetics with product inhibition have been used. He further assumed that quantity of phosphate used during the fermentation is proportional to tetracycline concentration as well as biomass concentration. The specific production rate depends on the activity of rate-limiting enzyme ATC-oxygenase . The model also includes rates of product and enzyme decomposition. Other models of tetracycline fermentation developed are that by Tarasova  and that by Bosnjak . The first model is an algebraic model of effect of pH on cell growth and tetracycline fermentation, while the second model is for repeated batch culture of (oxy) tetracycline. Both are unsuitable for simulation.
Materials and methods
Hybrid chemotaxis bublesort (H.C.B.)algorithm
(1) Find a suitable function to describe the feed rate profile. For example [u.sub.1] = a0 + a1 * t + a2 * [t.sup.2] with constraints that when V = [V.sub.max] or t = [t.sub.f] [u.sub.1] = 0.
(2) Initialize the (n) parameter values (a0, a1, a2 etc.), now not optimal.
(3) Start outermost loop, start (n-1) inner loop, sequentially, one loop for one parameter value (the parameter value of the outer loop remain constant while that in the inner loop changes.
(4) Do chemotaxis
(a) while in the inner loop add a random change to the parameter value
(b) Get the control variable (phosphate addition rate for this case) with current parameter values.
(c) Simulate the previous equation to get the relevant value of the state variables.
(d) Calculate profit from the profit function
(e) Compare profit with previous value of profit
(f) If there is no improvement, the old value of parameter is retained.
(g) If there is an increment the new value of parameter is retained and another random increment is added to the parameter
(5) Exit this loop after allotted number of iteration
(6) Activate immediate outer loop and add random increments to the second parameter, reenter inner loop and do all steps of 4) again
(7) Bublesort for maximum value of the profit functions out of all these possible combinations.
(8) When optimal value of a lower order profile is obtained, another loop to the program is added and the optimal profile of immediately higher order is explored.
The hybrid bublesort chemotaxis algorithm is a good compromise between the analytical and trial and error approach of optimization.
The profit function for the tetracycline optimization is as follows:
Profit = [V.sub.f] * [P.sub.f] * Selling price of tetracycline - [integral][u.sub.1] * cost price of phosphate - [integral] [u.sub.2] * cost price of soya-bean oil.--Downstream processing cost
Where [V.sub.f] = Final volume of the fermenter, [P.sub.f] = final tetracycline titer, [integral] [u.sub.1] = total phosphate added, [integral] [u.sub.2] = total soya-bean oil added, downstream processing costs are assumed to decrease exponentially with tetracycline titer.
The criticism of this profit function is that phosphate cost is insignificant in total process cost of producing tetracycline. However, this is an abstract rather than actual profit function and effect of addition of different concentration of phosphate will be manifested through change in the cell concentration and the titer of tetracycline also. Data was collected to obtain numerical values in the profit function and given in table 1.
Modified Votruba model  for simulation tetracycline production in Fed-Batch
The model equations may be as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
dE/dt = [k.sub.5] * tau - [k.sub.6] * E - [mu] * E (since the enzyme is an intracellular thing it does not get diluted due to volume increase of the reactor during fed-batch, but cell mass or sugar substrate get diluted )
dP/dt = [k.sub.3] * tau * X - [k.sub.4] * P - P/V * dV/dt
[S.sub.1] = substrate sugar concentration
[S.sub.2] = phosphate concentration
X = biomass concentration
P = product tetracycline concentration
E = the key enzyme concentration
[k.sub.1] - [k.sub.6] = rate constants
[mu] = specific growth rate
[Y.sub.x/s] = yield of cells on substrate
[u.sub.1] = phosphate addition rate
[u.sub.2] = soya-bean oil feed rate
V = volume of the fermentation broth
Table 2 gives the parameter values for simulation of the above model
The differential equations were integrated with MATLAB routines ode23 (based on third order Runge-Kutta) and ode15s a stiff solver based on Gear's method. The H.C.B algorithm was also coded in MATLAB.
Starting feed profiles
A good and biologically relevant starting profile is needed for a meaningful optimization. We have taken the values from the paper of Ross . Though, at present, we are experimentally determining the optimum range for phosphate concentration.
Table 3 gives concentration vs. time data for phosphate concentration were obtained from Ross' paper.
A simple material balance with assumption that no significant amount phosphate is being consumed by the cells can be made. Let [C.sub.i] is the feed concentration of phosphate, V = reactor volume and [C.sub.reactor] be the concentration in the reactor.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Now if we set [C.sub.i] = 30 mg/liter , V = 30 liter we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since these are just starting profile ,such simplification will not harm.
Now, if [u.sub.1] is taken as a first order profile [integral](a0 + a1 * t)dt = C after integration it becomes a0 * t + a1/2 * [t.sup.2] = C or b0 + b1 * t = C/1 now if a first order polynomial is fitted with time vs. concentration data of Table 3 with MATLAB routine of 'polyfit' we can extract the coefficients for flow rates by the following relationships a0 = b0, but a1 = 2b1 and so on. The detail relations between the concentration profile coefficients and flow rate coefficients have given in Table 4.
Finally we get, [u.sub.1] = 0.090793 liter/hr for a zero order (constant) flow = 0.167050 - 0,025014 * t liter/hr for a first order flow = 0.279067 - 0.146272 * t + 0.010929 * t^2 liter/hr for a second order flow = 0.475919 - 0.55254 * t + 0.142306 * t^2 - 0.0083 * t^3 liter/hr for a third order flow
Results and Discussions
The results indicate that a third order decreasing feed-rate profile is the best . We run two variants of second order optimization program. In the first variant a1 and a2 were varied in the same loop. In the second variant a0, a1, a2 was varied in three different loops. There is only a slight decrease in the value of [J.sub.max] (985 to 980). This slight change does not justify a ten fold increase in the computing time. Knowing this, we varied all the three parameters (a1, a2, a3) in a single loop for optimization of the third order profile. The results of optimization, the optimized value of flow coefficients and corresponding [J.sub.max], highest profit function value is given in table 5.
The values of the profit functions are only comparative and are not absolute. The chemotaxis algorithm without bubble-sort tend to forget the global maxima each time it re-enters the innermost loop. The results show our algorithm successfully clings to global maximum (Table 6).
We needed to compare the optimized fed-batch situation with the pure batch situation when the entire phosphate was added in the beginning of the fermentation (=0.0907925x15 hour =1.36 liters of solution with [C.sub.i] = 30 mg/liter, V = 30 liter), this corresponds to an initial concentration of (30x1.36)/31.36 = 1.3028 mg/liter). Table 5 shows that there is considerable improvement in total productivity with optimized fed batch from pure batch culture ( J = 1486 from J = 884), though total phosphate added are roughly same. However, fed batch shows less final tetracycline titer than from batch due to dilution effects (Figure 1)
The simulation of this type is often plagued with absurd increase in broth volume. This is due to errors in the starting profiles and not due to inherent defect of the algorithm. It can be easily tackled by putting a maximum volume constraints (When V = [V.sub.max], both [u.sub.1] = 0 and [u.sub.2] = 0). Figure 2 compares broth volumes between a pure batch and a fed-batch fermentation with third order (optimized) flow rate of phosphate.
Figure 3 shows the expected tetracycline titer time profile under third order (optimized ) flow rate of phosphate, while the figure 4 shows the concentration time profile of phosphate in the fermenter that is obtained under such a flow rate of phosphate.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
We have considered a fixed time problem. With volume /concentration constraints we can also consider time-free problems using the same algorithm.
[S.sub.1] = substrate sugar concentration gram/liter
[S.sub.2] = phosphate concentration milli-moles/liter
X = biomass concentration gram/liter
P = product tetracycline concentration milligram/liter
E = the key enzyme concentration
[k.sub.1] - [k.sub.6] = rate constants
[mu] = specific growth rate
[Y.sub.x/s] = yield of cells on substrate gram/gram
[u.sub.1] = phosphate feed rate litre/hour
[u.sub.2] = soya-bean oil feed rate litre/hour
V = volume of the fermentation broth litre
PBC = plasmid bearing cells
PFC = plasmid free cells
H.C.B. = hybrid chemotaxis bubble-sort algorithm
I.P.T.G. = iso-propyl-[beta]-D-thio-galactopyranoside
P.A.A. = phenyl acetic acid
J = value of the profit function.
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(1) S. S. Maitra *, (1) Saurabh Chugh and (2) Neha Singh
(1) School of Biotechnology, Jawaharlal Nehru University, New Delhi -67 Pin -110067 India E-mail: email@example.com
(2) Department of Biochemistry, J.C. Bose Institute of Life Sciences, Bundelkhand University, Jhansi, Uttar Pradesh Pin-284128 India
* Corresponding Author
Table 1: Cost and sale prices of various items important for getting a numerical value of the profit function. sale price Tetracycline Rs. 1.2 per gram cost price Phosphate Rs. 0.80 per gram cost price Soybean oil Rs. 0.083 per gram Downstream processing cost function=2*exp(-final conc. of tetracycline). Table 2: Parameters for model simulation. Parameter Dimension Modified Values [k.sub.1] [hr.sup.-1] 0.012 [k.sub.2] [hr.sup.-1] 0.054 [k.sub.3] [hr.sup.-1] 0.616 [k.sub.4] [hr.sup.-1] 0.02 [k.sub.5] [hr.sup.-1] 10.8 [k.sub.6] [hr.sup.-1] 0.096 [K.sub.11] mg/litre 200 [K.sub.12] g/litre 0.058 [K.sub.13] m.mol./litre 20.53 [K.sub.s] g/litre 0.5 [1/Y.sub.x/s] g/g 5.87 Table 3: Concentration of phosphate vs. time data for obtaining starting profile. Fermentation Molar conc. in Phosphate concentration/time Time in hours m.Mols/liter concentration in (x) (t) milligrams/liter 1 1.4 0.245 0.245 2 1.1 0.193 0.0965 5 0.6 0.105 0.021 15 0.0571 0.01 0.00067 120 0 0 Table 4: Coefficients of feed profiles obtained from concentration profile data. zero order 0.090793 1st.order b0=0.167050 b1=-0.012507 a0=0.167060 a1=2 * b1= -0.025014 2nd.order b0=0.279067 b1=-0.073136 b2=0.003643 a0=0.279067 a1=2b1= a2=3 * b2= -0.146272 0.010929 3rd.order b0=0.475919 b1=-0.27627 b2=0.047436 b3=-0.002075 a0=0.475919 a1=2b1= a2=3 * b2= a3=4 * b3= -0.55254 0.142306 -0.0083 Table 5: Results of optimization. order value of a0opt a1opt a2opt a3opt [J.sub.max] pure batch 884 First order 898.98 0.2023 -0.0188 Second 985.61 0.3445 -0.1098 0.0136 order(1) Second 980.71 0.3454 -0.1119 0.0134 order(2) Third order 1486.7 0.4198 -0.4226 0.1679 -0.0064 order final tetracycline titer (mg/liter) pure batch First order 2.4276 Second 2.2884 order(1) Second 2.3524 order(2) Third order 2.0846 Table 6: Data establishes efficacy of H.C.B algorithm. a0 a1 J(i,j) Jmax a0opt a1opt 0.1410 -0.0249 889.5459 889.5459 0.1410 -0.0249 0.1410 -0.0239 889.7574 889.7574 0.1410 -0.0239 0.1410 -0.0246 889.6163 889.7574 0.1410 -0.0239 0.1410 -0.0264 889.2369 889.7574 0.1410 -0.0239 0.1410 -0.0197 890.9926 890.9926 0.1410 -0.0197 * 0.1410 -0.0227 890.0519 890.9926 0.1410 -0.0197 0.1410 -0.0264 889.2285 890.9926 0.1410 -0.0197 0.1410 -0.0209 890.5357 890.9926 0.1410 -0.0197 0.1410 -0.0265 889.2165 890.9926 0.1410 -0.0197 0.1410 -0.0218 890.2823 890.9926 0.1410 -0.0197 0.1410 -0.0261 889.3054 890.9926 0.1410 -0.0197 0.1410 -0.0251 889.4967 890.9926 0.1410 -0.0197 0.1410 -0.0246 889.6137 890.9926 0.1410 -0.0197 0.1410 -0.0255 889.4115 890.9926 0.1410 -0.0197 0.1410 -0.0305 888.5415 890.9926 0.1410 -0.0197 0.1410 -0.0230 889.9557 890.9926 0.1410 -0.0197 0.1410 -0.0238 889.7790 890.9926 0.1410 -0.0197 0.1410 -0.0226 890.0747 890.9926 0.1410 -0.0197 0.1410 -0.0239 889.7443 890.9926 0.1410 -0.0197 0.1492 -0.0237 890.4773 890.9926 0.1410 -0.0197 0.1492 -0.0251 890.1221 890.9926 0.1410 -0.0197 0.1492 -0.0278 889.5378 890.9926 0.1410 -0.0197 0.1492 -0.0226 890.7817 890.9926 0.1410 -0.0197 0.1492 -0.0202 891.5869 891.5869 0.1492 -0.0202 * 0.1492 -0.0243 890.3098 891.5869 0.1492 -0.0202 0.1492 -0.0246 890.2513 891.5869 0.1492 -0.0202 * 0.1492 -0.0189 892.0358 892.0358 0.1492 -0.0189 0.1492 -0.0271 889.7238 % 892.0358 0.1492 -0.0189 0.1492 -0.0278 889.5327 892.0358 0.1492 -0.0189 0.1492 -0.0223 890.8739 892.0358 0.1492 -0.0189 0.1492 -0.0220 890.9668 892.0358 0.1492 -0.0189 0.1492 -0.0276 889.6014 892.0358 0.1492 -0.0189 0.1492 -0.0286 889.4052 892.0358 0.1492 -0.0189 0.1492 -0.0311 888.9812 892.0358 0.1492 -0.0189 0.1492 -0.0247 890.2429 892.0358 0.1492 -0.0189 0.1492 -0.0301 889.1440 892.0358 0.1492 -0.0189 0.1492 -0.0244 890.2981 892.0358 0.1492 -0.0189 * (semi) global maxima % ordinary chemotaxis forgets the semi global maxima on loop re-entry but our H.C.B. algorithm successfully clings to it (data shown for optimization of first order profile, first few rows only)
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|Author:||Maitra, S.S.; Chugh, Saurabh; Singh, Neha|
|Publication:||International Journal of Biotechnology & Biochemistry|
|Date:||May 1, 2010|
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