An Approach Using Bezier Curves to Control pH and Decrease Enzyme Inactivity

Biosynthesis may require an active enzyme concentration in a cell-culture growth medium. Reduced biosynthesis capability caused by an inactive enzyme can be an economical concern. Controlling the inactivity of pH-sensitive enzymes can help maintain, or possibly increase, a stable microbial growth environment, thereby preventing a shift in the biosynthesis capabilities of the process harvest.

The biosynthesis of pharmaceutical drug products begins by evaluating charts from the product development laboratory that show a drug's production capability relative to pH, time, substrate concentration, enzyme concentration, and so forth. These informational components were the basis for the "Enzyme Wyeth/RoviSys Controller" (EWrC, RoviSys, Aurora, OH) discussed in this article. The algorithms for the EWrC have been developed and tested on a personal computer. The human–machine interface has not yet been developed.

Figure 1: This reaction mechanism involves six kinetic constants, making it difficult to analyze.

To understand how the information provided in product development laboratory charts can fundamentally improve the approach to process control, it is necessary to briefly review the analytical methods used to evaluate biochemical reactions. Biological cellular growth mediums that make up enzyme-catalyzed reactions are difficult to work with analytically for several reasons (1):

• To achieve lower activation energies for a reaction, the concentration of the enzymes must be extremely low compared with the concentration of the substrate. These low concentrations and the complexity of enzyme–substrate binding make it difficult to determine what the enzyme is actually doing.

• Complex kinetic constants of large enzymes cannot be determined by steady-state experiments. Often, at least six kinetic constants k are involved in a step-reaction sequence for producing a particular drug (see Figure 1).

• Molecular dissociation in these biological media is difficult to determine.

• Conventional steady-state equilibrium methods are not linearly applicable to enzymes.

• Activities of many enzymes vary with pH in the same way that simple acids and bases ionize. In contrast, certain enzymes involved with sequential reactions associated with microorganism's growth kinetics dealing with biosynthetic pathways are very complex, diverse, and huge (molecular weights can be as high as one million daltons).

• Enzyme reaction velocities are subject to activity inhibition caused by reactive competition from other components in the medium or even the substrate.

Standard state and relaxation-method analysis. Subsequently, physical biochemists have defined a standard state in which all of the activities and concentrations normally used to analyze reaction kinetics are not required. Hammes developed an analytical method known as the relaxation method to study fast chemical reaction components at equilibrium. This method can be briefly stated as follows:

A chemical system at equilibrium is suddenly perturbed in such a way that it finds itself out of equilibrium under the new conditions. It then relaxes to a state of equilibrium under the new conditions, and the rate of relaxation is governed by the rate constants and mechanism of the process (1).

Relaxation-method analysis and steady-state theory describe the physical model of an enzyme-catalyzed reaction shown in Figure 1, with the following precepts:

• At equilibrium, the rates of forward and back reactions are the same; that is, k₁ = k_–1, k₂ = k_–2, and k₃ = k_–3.

• Only the ratio of rate constants, not individual rate constants, can be determined.

• Many reaction properties undergo constant and rapid fluctuations about some average or median state. Any long-term measurements provide the same results if the system is truly at equilibrium.

• Relaxation kinetics provide simple, first-order exponential time results regardless of the number of molecules involved as either reactants or products.

• After measurement analysis, concentrations charted over time fall on a straight line.

• Steady state is an approximation as the substrate is gradually depleted. If the rate measurements are restricted to a short-time interval over which the concentration of the substrate does not greatly change, however, then it is a very good approximation.

Departing from conventional control strategies, the principle objective of the EWrC is to maintain the equilibrium by controlling the reaction perturbation defined by the product-development laboratory charts and supported technically by the relaxation methods' physical model for the enzyme-catalyzed reaction of interest.

Transition-state theory. The transition-state theory, supported by the Hammond postulate for Gibbs free energy of mathematical maximums and minimums for reaction growth diagrams, is used to divide the laboratory charts into various reactant intermediate states (see Figure 2) (2). The transition-state theory for enzyme-catalyzed chemical reactions states the following:

Figure 2: Transition-state theory.

• In a reaction-growth diagram, transition states occur at mathematical maximums and intermediates occur at mathematical minimums.

• At mathematical maximums, chemical bonds are in the process of being made and broken.

• At mathematical minimums, intermediates have chemical bonds fully formed.

• The reaction growth diagram plots the energy of the reagents as the reaction proceeds. Both growth and pH will be used in this article.

In addition, the following portions of the Hammond postulate are of interest:

• If there is an unstable intermediate in the chemical reaction pathway, the transition state for the reaction will resemble the structure of the intermediate. The reasoning is that the unstable intermediate will be a mathematical maximum in the reaction-coordinate diagram.

• Gibbs free energy curves for a substrate and product intersect on the reaction coordinate at the position of the transition state. A change of structure away from the seat of reaction destabilizes it, displacing its energy curve higher, and the point of intersection moves closer to the substrate.

The importance of transition-state theory is that it relates the rate of a reaction to the difference in Gibbs free energy between the transition state and the ground state.

EWrC development

Selection of the nonlinear control strategy has been approached theoretically (3). As observed by Antje Kann,

Models are often developed with knowledge of theory and algorithms, but who may not be intimately familiar with the users' situations. Modeling choices made are often guided by practical concerns such as solvability, run-time issues, and ease of implementation in code. However, these choices may impair the ease with which a model can be applied to solve the users' problems (4).

Following this train of thought, a fresh approach in the development of the experimental EWrC was taken. The product-development laboratory charts were the best starting point. Well-documented analysis was available for all the process dynamics, measured variables, responses, and unanswerable process questions for the drug of interest. Classical multivariable, nonlinear control strategies provided well-documented information. In addition, a good curve-generating engine would help convert data from the laboratory charts into useful information.

Bezier curves. Parametric equations allow curves to have the unusual ability to move straight up or even curve back and cross themselves (see Figure 3). Cubic Bezier curves belong to the family of parametric curve equations. Cubic Bezier curves were selected as the first set of parametric curves to use because they do not need a tangent vector. Bezier curves are characterized by defining four coordinate points (origin, destination, values of x, and values of y), one linear variable coordinate point t, and two polynomial equations. The two polynomial equations are used to calculate the values of x and y between the increasing variable t.

Figure 3: Example of a parametric curve crossing itself twice.

As a simple example for the development of a cubic Bezier curve (5), suppose two polynomial equations define the coordinate points on the curve. Both are evaluated for an arbitrary number of values of t. In this example t is incremented according to:

t = t + 0.02, with t ≥ 1

which generates 50 values of t. As increasing values for t are provided to the equations, the point defined by x(t), y(t) moves from the origin to the destination according to:

x(t) = axt³ + bxt² + cxt + x₀

x₁ = x₀ + cx/3

x₂ = x₁ + (bx + cx)/3

x₃ = x₀ + ax + bx + cx

y(t) = ayt³ + byt² + cyt + y₀

y₁ = y₀ + cy/3

y₂ = y₁ + (by + cy)/3

y₃ = y₀ + ay + by + cy

The values of the coefficients a, b, and c can be determined with the following equations:

Σcx = 3(x₁ – x₀)

bx = 3(x₂ – x₁) – cx

ax = x₃ – x₀ – cx – bx

Selecting four control cubic coordinates at random:

origin: xp₀ = 3.5 and yp₀ = 0

control point 1: xp₁ =3.5 and yp₁ = 1

control point 2: xp₂ = 5 and yp₂ = 6

destination: xp₃ = 6 and yp₃ = 3

These four coordinates evaluate to the coefficients: cx = 0, cy = 3, bx = 4.5, by = 12, ax = –2, ay = –12. The first coordinate pair at t = 0.02, provides x(0.02) = 3.50 and y(0.02) = 0.06. Similarly, at t = 0.04, x(0.04) = 3.50 and y(0.04) = 0.14 (similarly for t = 0.06, etc.). Figure 4 shows the dramatic difference between the results of two selections for control coordinates.

Figure 4: Two examples of cubic Bezier curves. Two different control coordinates were selected for (a) and (b). By using four Cartesian coordinates instead of two to define a point on a curve, calculating correctly dependent variables based upon nonlinear high-gain independent variables is deterministic rather than infinity.

Controller types. Using loosely coupled data from laboratory charts and borrowing concepts from classical controllers with multivariable, nonlinear capabilities generally used for pH regulation produced the following filtered list of controller types:

• Endpoint control (EpC). The strength of this type of controller is its use of stoichiometrics. The stoichometrics associated with enzyme–substrate reactions occurring in cell-culture growth media are highly complex, both physically and chemically. The EWrC will use measured variables as an indication value and associated other parameters before determining the value of the final control element.

• Gain compensation control (GCC). This type of controller can function with nonlinear control measurements in real-time. The EWrC will use this concept by giving Bezier capability to the laboratory charts for pH, time, and product growth rate and adjusting the known process nonlinearities to predictable or reduced or, in some cases, eliminated controller response.

• Adaptive control (AC). Controller mode values change with the dynamics of measured variables. EWrC will use this concept by giving Bezier capability to the laboratory charts for pH, time, and product growth rate and adjusting the controller output to known measurement dynamics. The controller and process will be self-regulating.

• Model predictable control (MPC). This can be used to predict near-future variable values on the basis of a model developed by pseudo random process variation studies. The EWrC will use the product development laboratory charts as the basis for the controller algorithm and provide a very definitive model.

Summary of information requirements. The EWrC now can be fully developed using the following information about the drug's properties, theories, methods, control strategies, and states:

• laboratory informational charts developed for the drug of interest;

• physical biochemical model precepts by Hammes' relaxation method;

• transition-state theory with the Hammond postulate;

• steady-state dynamics;

• pseudo steady state;

• EpC concept of stoichiometrics;

• GCC concept for nonlinear measurements;

• AC concept of mode changes with measured dynamics;

• MPC concept of near-future predictions for process variables.

The following detail requirements will serve as the basis for the controller:

• Three laboratory determined sets of data for product-growth rate versus pH, growth rate versus time, and growth rate per pH versus time will be experimentally developed.

• The three sets of laboratory data for product-growth rate will be formulated with cubic Bezier control.

• EpC has components that control the proportional feeding (stoichiometric) of reactive agents in either correct or excess proportions. The EWrC will calculate the standard deviation of the product-development laboratory's experimental data and determine whether the batch has either a correct or excess proportion of reactive agents.

• GCC has a control component that varies the loop gain compensation with nonlinear compositions. The EWrC will calculate the required loop gain on the basis of the time interval from the state of the batch, pH, and estimated growth rate.

• AC has components that change the control modes on the basis of the EWrC's active control mode. The EWrC will change the slave proportion-integral-derivative (PID) controller modes and values on the basis of set-point limits (based on the standard deviation of the active control mode), integral set by the active median value, derivative set by the active estimated growth rate, and the automatic–manual– remotecascade (operator selection).

• MPC has a control-horizon component that uses the model developed for the various coupled control variables to estimate a future value. The growth-rate curve determined by experimental methods in the laboratory will be the model.

• Laboratory-determined data-point coordinates would be formulated with cubic Bezier control.

The following measurements will be made and their values compared with the desired values:

• time batch started;

• real-time clock reset to zero at the time the batch started;

• pH;

• gain (pH desired at time xversus pH actual at time x; or growth rate at pH actual at time xversus growth rate estimated at time x).

Functional specifications for the EWrC (see Figure 5) to be developed are as follows:

• pH measured and displayed on a screen with historical capability;

• pH measured data given cubic Bezier capability and displayed on a screen with historical capability;

• chemical transition states, each displayed on a screen;

• chemical-transition states data given cubic Bezier capability;

• chemical intermediates, each displayed on a screen;

• chemical intermediates data given cubic Bezier capability;

• real-time growth curve for microorganism of interest and displayed on two screens, each with historical capability;

• growth-rate curve data will be linked to either a chemical transition state or chemical intermediate;

• estimated growth rate curve screens will have real-time clock displayed linearly;

• pH measured data will be linked to each growth rate curve screen;

• pH measured data gain will be linked to each growth rate curve screen;

• cubic Bezier control capability for all data sets with historical capability.

Table I: EWrC detail specifications.

Detail specifications for the EWrC are shown in Table I and Figure 5. A block diagram for the EWrC is shown in Figure 6. For clarity, alarms and emergency anomalies are not shown.

Figure 5: Example of the human–machine interface for the EWrC (see Table I for descriptions of each screen).

Control scenarios

To demonstrate the EWrC, the following are four simulated control scenarios for making a batch with the same set of growth rate curves.

Figure 6: Example of a block diagram for the EWrC.

Scenario 1. The "pH dominate" (pHdom) mode will continuously challenge the current pH measurement and test whether it is possible to increase the pH value ahead of the time period determined by the laboratory data (see Figure 7, scenario 1, screens A and B). Referring to screen B, with the EWrC in pHdom mode, the gain increases at almost a linear slope (see screen B). The desired pH ramp is requested to increase at a much faster rate than normal. The growth rate curve is unaffected in this mode.

Figure 7: Examples of four different control scenarios of the human–machine interface for (a) pH dominate, (b) pH normal, (c) rate dominate, and (d) rate normal modes.

EWrC_output = IF(pH_{desired at time x}/pH_{actual at time x})

within Standard Deviation THEN pH_current(pH_{desired at time (x + 1)}/pH_{estimated at time (x + 1)}) ELSE pH_current(pH_{median at time x})

Scenario 2. The "pH normal" (pHn) mode does not challenge either the predicted product growth time or the pH of the laboratory data (see Figure 7, scenario 2, screens A and B). Referring to screen B, with the EWrC in pHn mode, the gain tracks the growth-rate curve. The growth-rate curve is unaffected in this mode.

EWrC_output = IF(pH_{desired at time x}/pH_{actual at time x})

within Standard Deviation THEN pH_current(pH_{desired at time x} /pH_{actual at time x}) ELSE pH_current(pH_{median at time x})

Scenario 3. The "growth rate dominant" (Rdom) mode will continuously challenge the current pH measurement and tests if it is possible to increase the pH value ahead the of the time period determined by the laboratory data (see Figure 7, scenario 3, screens A and B). Referring to Screen B, with the EWrC in Rdom mode, the estimated growth rate and time interval for pH changes is used in combination to determine the gain output. The growth rate desired increases rapidly and then falls into a constant decline. A general low gain and constant gain value is determined necessary for most of the batch process. The pH curve is unaffected in this mode.

EWrC_output = IF(Rate_{estimated at time x}/Rate_{desired at pH actual at time x})

within Standard Deviation THEN Rate_current(Rate_{estimated at time (x + 1)}/Rate_{desired at pH estimated at time (x + 1)}) ELSE Rate_current(Rate_{estimated at pH median at time x})

Scenario 4. The "growth rate normal" (Raten) mode does not challenge either the predicated product growth time or the pH of the laboratory data (see Figure 7, scenario 4, screens A and B). Referring to Screen B, with the EWrC in Raten mode, the estimated growth rate is used to determine the gain output. A very large gain is determined necessary at first and then falls off rapidly to a constant value until the end of the batch process. The pH curve is unaffected in this mode.

EWrC_output = IF(Rate_{estimated at time x}/Rate_{desired at pH actual at time x})

within Standard Deviation THEN Rate_current(Rate_{estimated at time x}/Rate_{desired at pHactual at time x}) ELSE Rate_current(Rate_{estimated at pH median at time x})

Conclusion

An experimental EWrC has been created and debugged to demonstrate and prove the algorithm on a personal computer (see Figure 8). The human–machine interface has not been created and the controller has not been tested in an actual bioprocess.

Figure 8: Example of a screen shot of the experimental EWrC.

The potenital benefits to product quality, especially for injectable drugs, are the best feature of the EWrC. Product quality associated with pharmaceutical bioprocessing is assessed in terms of the finished product's physical and chemical properties. The EWrC and its control functionality are completely based upon the physical biochemical properties developed in the product development laboratory for the product of interest.

The limitations of the EWrC lie in the statements of physical biochemists listed at the beginning of the article. Reaction intermediates produced in situ accompanying the large volumes used in commercial batches could alter relaxation methods responses markedly. In addition, the forward and back rates of reaction can be affected nonlinearly with perturbations.

The operational and controls benefits of the EWrC functionality depend on the specific physical biochemical properties of the drug of interest. On the basis of the product development laboratory charts, there should be less control transitions because the addition of reagents should follow the relaxation method principles more closely. In addition, the system is less affected by concentration anomalies in reagents; control with either biologics or difficult or latent process measurements is possible; and the system is self-optimizing in pH or rate-dominate control modes.

Production economics benefits can be achieved from the various EWrC modes. For the pH or rate-dominate modes, reagent addition should be at an optimum rate, thus increasing product concentration, decreasing batch perturbations, and decreasing batch-production time intervals. In addition, operating close to equilibrium should optimize utilities and reagent usage. For the pH mode, reagent concentration anomalies should not perturb equilibrium because time-to-pH values is used as a parameter. In addition, operating close to equilibrium should optimize utilities and reagent usage. Finally, the rate mode applies primarily for reactions experiencing enzyme-substrate binding that cause latent pH measurements, and operating close to equilibrium should optimize utilities and reagent usage.

Michael J. Needham is a validation engineer at RoviSys, 1455 Danner Drive, Aurora, OH 44202, needham_michael@hotmail.com

References

1. I. Tinoco, Jr., K. Sauer, and J.C. Wang, Physical Chemistry Principles and Applications in Biological Sciences (Prentice-Hall, Englewood Cliffs, NJ, 2d ed., 1985).

2. Alan Fersht, Structure and Mechanism in Protein Science (W. H. Freeman and Company, New York, NY, 3d ed., 2000).

3. Instrument Society of America, Standards and Practices for Instrumentation (ISA, 6th ed., Research Triangle Park, NC, 1980).

4. Antje Kann, president, Analytics Interactive (San Francisco, CA, www.analyticsinteractive.com).

5. "Bezier curves," http://www.math.ubc.ca/~cass/gfx/bezier.html, accessed, Jan. 16, 2006.