Dav M. Shindell, Carl Magagnosc, and Daniel L. Purich

I Teaching with an Interactive Reaction Unlverslty of Cal forma Santa Barbara. 93106 I Kinetics Simulator

Simulation techniques constitute an almost inescapable tool for analyzing the kinetics of complex chemical systems ( I ) . Both analog and digital methods have been commonly employed to predict the time dependence of various reaction systems (many of which do not have general mathematical solutions), to validate relationships between models and oh- served svstems, and to estimate the magnitude of rate con- stants that cannnt he mw~sured d~rcct l i . In addition Lo its truditionnl use in thc lahoratory,simulatirnn methodsare vrrv itttractivr for the teaching c ~ f chemical and enzyme kinetics c,,uries. There, they rnn he used to demonstrate various hasic kinetic princ~ples, illuitrare the use of a m m o n assumptions (such its the stendv state and equilihriunl assumptionsl, and provtde students with a feeling for the mngnitudes 111 rate constants in particuli~r systems.

Kinrttc simulnti~m has evolved cnnsiderahly since early mechantrill diffrnwtial nnalyrers wrre first applied by Chance , 2 ) w e r 30 ) w r s sgu to the simulation of simple rnzyme sys- tems. Ik:lt.ctronic analog computers have sincr heen routinely usrd to model small svntenis, hut the circuitry ol'nn analog computer can quickly become too complicated to program as the number of branched pathways or intermediate species increases.

Use of high speed digital computers has been a great ad- vance for those interested in kinetics. as it has simnlified the task of programming large schemes (1.3-5). In the simplest form, these packages of computational routines attempt to directly replace the analog computer with digital algorithms, requiring a circuit diagram analogous to the one needed on an analog computer. Programming of a particular scheme is ac- complished hy inputting parameters and connections in tabular form. More advanced packages (such as the IBM CSMP and the CDC MIMIC) employ a Fortran-like structure for the inputting of the differential equations describing a articular mechanism. In all the above cases, the user has the iesponsibility to translate the chemical mechanism into a form usable by the computer and to select simulation parameters consistent with the required accuracy. Recently, Edelsen (6) has employed a translator to take pseudochemical notation innut and automaticallv nreoare a data table for a similar . . . integration program.

In considerinr! the need for an imoroved simulator with a much s i m l h pr&ncc81 for use in the ciassnmm and laboratory, we drfined t hr fcdlowing ;IS essential featurrs: I IJ a turtl-key deviw with t h ~ simplrst possible operation, 121 useof natural chemical I;rtiau,xy and nutatton, (3) elimination (nf the inter- mediate steps of -kiting differential equations and translating them into an analog circuit or a digital program, (4) an accu- racv much imoroved over the tvoical small analor! comwter, - . (5)provision ;or the user to interact with the simulator and ranidlv modifv innut narameters. (6) automatic selection of . . . . . cimulntion parameters 1 8 , ensure rccurste and rapid simula- rwni u ith on internal ehtimatim oferror. and (71 mdes t cost (88-10,000) and size (desktop). Together, these features specify an inexpensive unit for student use without requiring any programming or computer knowledge, especially impor- tant in teaching introductory or survey courses. The reaction kinetics simulator shown pictorially in Figure 1A meets all the above criteria. This digital simulator, of course, constitutes a compromise hetweenthe performance of electronic analog computers and large digital computers. The simulator is

limited to about 15 uni- or hi-molecular reactions with as many 11s 20 chemical species, pr~vid6.d the scheme is not particularly stiff. Certainly, this is much less thnn other, more elahorate systems such as the HKI.L(.'HEM compiler 161. Yet. fa,r most phvsirul.organic and general chrmical kinetics work, the number of elementar!. atens falls a,ithin the ranrtfof this " . - system.

In this report, we describe the teaching applications of a micro computer-based reaction kinetics simulator meeting the above requirements and possessing other convenient features (7). In particular, we illustrate the utility of the simulator in teaching chemical principles by use of selected examples.

Rtnrllo* I s n l / l c * l / O * I *BBOIY,* , IY 11lmrTl01i:

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s*,n I e L I r I I 81 I l O S l e L L l "@l. lE* 1,1111 l l ,"S" I T , ""E* ,Hi i<*mr li

# e l / IO*sm"re m " n E , E L " EW.WF0

8 > = 18 ' i . r E W l E l W.lt CI1"81*"1 YI)LYI: I O i . t%* sri(Crl0"

I,,,,, *L I D U C I Y I I 1 L T I O Y I

I", ~ m,es ,",,,(I, m , m " l 7 ~ l l ~ * I I* ::: i g /I*,,% rONs,Plll,T "I," il*n LOnrillXrr

Figure 1A. Configuration of lhe miaomrnputer-based reacfion kinetics simulator complete with graphics terminal, microcomputer, and permanent copy unit. B: Illustration of simulator input format with abbreviated instruclions, and C: Plot 01 A. B, and C concentration versus time for a series first-order reaction. The initial concentration of A was 1 Mwith [E l ] and [C] initially zero. Rate Constants of k , = 10 s-' and k2 = 1 5-I were used. Time required to simulate was 8 S.

708 / Journal of Chemical Education

Teachlng Examples To reinforce basic kinetic principles in teaching, it is often

desirable to consider a number of exnerimental situations and typical reaction mechanisms. Of courie, this can be ;ircum- plished hy mathematical derivations in thr classm~m, but the time exprnded on a particular exiunple is nett always suited to the variation in stud en^ interest ;and intuition. Likewise, while literature references to actual chemical systems con- stitutv a valuntde exposure for students to experimental dnta, nuvices are often confused hy the usr of nonstandard notation or format. IJse of an analog computer to achieve this exposure is limited by the time that astudmt must spend pntrhinganrl debugging analog circuits. The usr of simulativn packages such ns CSMP is similarlv limited bv the time snent writing and debugging the digital program. 1; contrast, we have founi that the micro~rocessor-based simulator can become a now- erful adjunct tb classwork, and that its interactive features permit learning at a speed adjusted to the student's own ca- pacity and interest. For the beginner, less than an hour's in- vestment can provide an exposure to four or five of the cases considered below.

Case I. Series First-Order Reactions There are a number of examples of processes which are

described by the following mechanism k, k~

A-B+C (1)

Sequential radioactive decay of unstable nuclei, some irre- versible metabolic processes, drug buildup and clearance in a particular cell compartment, and the thermal decomposition of acetone offer pertinent examples. The maximum concen- tration of B, of course, depends upon the relative magnitude of the rateconstants. Tosimulate this scheme, the information is tvned on the s ra~h ics terminal (seen in Fie. 1A) in the for- - . matshown in ~yg&e 1B. The undkrlined items represent in- formation nrovided hv the user in resnonse to format state- ments which appearas each step iscompleted. Since the kevboard emnloved does not have arrows. - and e are ren- resented by 5 and =.Any or all of the reactants or can he plotted in a linear, loaarithmic, or derivative (rate) format.~~bsorbance can also be plotted if the necessary ex- tinction coefficients are known. (Other details of the simulator will he described in a later communication.) In Figure 1C the display of one series first-order process is presented. Since the capacity of the microcomputer is relatively large, one can do multiple simulations simultaneously by using different sym- bols (i.e., A - B - C, D - E - F, etc.). This was done in

TlME (SEC) Figure 2. Plot of the concentration of the intermediate B for various values of k, as shown in the graph. Other parameters are listed in the legend to Fig. 1 C. Simulation time was 28 s.

Figure 2 to obtain a record of the effect of the size of the second rate constant on the buildup of the intermediate.

Case 11. Reduction of Bimolecular Kinetics to the Pseudo- First-Order Aooroximation . .

Another prominent kinetic principle, which is extremely usrful ex~erimentallv. is the reduction of the order ofa reac- tion. FO; bimolecul& processes which are first-order with resnect to each reactant, A and B, the integrated bimolecular rate equation is

where a and b represent respective initial concentrations, and x represents the decrease in the concentration of either reactant at a given time.. Students often are given statements such as the one in Frust and I'earson ! R I : " I f conditions for a given reaction are such that one or more of the concentration factors are constant or nearly constant during a 'run,' these factors may be included in the constant k." With this, the student can mathematicdlv reduce em. (2) bv assumine that ( b - x)lb is essentially unity and indudin; the (b - ai term into the rate constant. Nonetheless. there is alwavs the question of how large [B] must he in piactice to give p8eudo- first-order kinetics with respect to component A. Such acase is considered in Figure 3. When [A] = [B], the deviation from first-order kinetics becomes more sizable as the reaction progresses. However, as shown in the inset, a ten-fold excess of [BI over [A] gives a reasonable pseudo-first-order approx- imation. It is also noteworthv that do t s of In IAl versus time . . somewhat obscure deviations from pseudo-first-order con- ditions since such lots look virtuallv linear when IB1 2 31A1. The important here is that thestudeut can gkt afeeifdr the reauired maenitudes necessarv bv first hand exnerience . - with t i e simula~ons.

Case 111. Enzyme Kinetics The kinetic nronerties of enzvmes are often of interest to . .

students who are fascinated by the rapidity of enzymic ca- talvsis. We mav consider the following to he a minimal - mechanism consistent with experimental evidence

E + S e X t Y r E + P (3) / bond \

breaking .terJ )

Here, X and Y are complexes of an enzyme with the substrate and product, respectively. In such cases, one can explore the various nhases of an enzvme catalvzed Drocess: re-steady state, steady-state, and near equilibknn.'~hese are shown in Figures 4A, B, and C. Simulation is especially useful here since

o l 1 0 0.1 a2 0.3 0.4 0.5

TlME ( M I N I

Figure 3. Plot of the concentration of A versus time for the bimolecular reaction A + B - C. Comparison is made between the case where [A] = [B] = 1 M and the eouivalemfirst-order curve. In the inset. IBI was set initially at 10 X [A]. . ~

For both simulations, the bimolecular rate constant was 10K' min-'. Simulation time was 4 5.

Volume 55, Number 11, November 1978 / 709

0 MI o m 001 0.w 005 TlME ISEC)

l5 scheme, and various values of rate constants were chosen to fit the model to the data. As shown in Figure 5, a reasonable

p value can he quickly identified, 1 and the example illustrates the U e x i h i l i t y of the simulator in - 2 estimating rate constants. It

should be noted that the size of the crossbars on the data points may he adjusted tore-

o 20 10 60 m 100 flect the experimental error. In TIWE ISEC)

C addition, one may overlay each trial simulation on the display screen for best comparison of sequential simulations as has been done in the figure. I t may he nointed out that the fit is best in the lower region loo'? time <6 hr). A much - . - E better fit over the entire reac-

$4 tion course can he accom- plished using a more realistic mechanism involving inter- mediate complex formation.

0 !u: TIME D ISEC) student This is not can shown rapidly here, hut survey the such cases with the simula-

Figure 4. Plots of the enzyme catalyzed reaction E + S rt X e Y - E + Pin its pre-steady state (A), steady state (El, and tor. approach to equilibrium (C) phases. All reactants and products are plotted. [El = lpM,[S] = 0.15mM. k+, = lo6 W'. k-, = 10 s-'. kt2 = 9 S-', kLp = 1 5-', k + ~ = 10 5-'. kL3 = lo5 M1 J. me Michaelis-Menten mechanism is illustrated Concluding Remarks 14Das a l imith case of the reaction in 148). Curves are sirnilarlv labeled. krr was chanaed to 1000 s-1. Simulation times This reoort has concentra- . . . . . - were 8 s: 196 s; 554 s. and 8 s. respectively.

there is no eeneral solution to the differential eauations (9). ~otewor thy is the nonlinear rise and fall of the first inter- mediate nrior to the steadv-state (Fig. 4A). Likewise, the simulatiok in Figures 4B and 4C show-that the steady state can persist for some time, justifying many of the steady state measurements made over the 10-100-s period following mixing of E and S (10). Another valuable concept illustrated hv such simulations is that X and Y maintain finite levels at e&dihrium reminding the student that the chemical species continue to shuttle forth and hack during eauilibrium. This is an important point when one wishes to iitrdduce topics such as isotope exchanee a t eauilibrium (10).

in all^, one may also show that the two-intermediate case can reduce toessentially one intvrmediat~ (the so-called Mi- chaelis-Menten mechanism,. This is shown in Figure iD, where the ?( -. Y interconversion has heen increased sub- stitntially, mnking [?() small relative to IYJ. Moreover, the process cakes on the properties uf a fast pre-equilibrium for- mation of (YI prior to the slow formation of [ E l and [PI.

Case IV. Autocatalysis There are many processes which involve autocatalysis, and

these are especially common in biochemistry. Blood clotting and activation of proteolytic enzymes are two prominent ex- amples. Here, the basic mechanism can be written in its sim- plest form

where A is the inactive form and B is the catalyst necessary for its activation to yield more catalyst. A classical example of autocatalysis was examined for the case of trypsin attack on trypsinogen by Kunitz and Northrup (11). The former is an active enzyme used in digestion of proteins while the latter is the inactive form discharged via the pancreatic duct into the small intestine. Upon entry into the intestine, trypsinogen is converted to the active form (trypsin) in an autocatalytic fashion. The data from the Kunitz and Northrup experiment were entered into the simulator along with the above kinetic

- ted on only a few kinetic schemes to illustrate the in- structional value of the micro-

computer-based simulator. For most solution studies of chemical reactions (no provision has presently been made for the treatment of termolecular processes sometimes observed in gas-phase kinetic prohlems), such miniature dedicated computer systems will doubtlessly play an increasing role as an adjunct to lecture material, leading to the treatment of a wider variety of mechanisms in the classroom. Certainly, the ease of operation will not only encourage the instructor to develop specific examples that are tailored to particular coursework, but the student will also tend to explore more model systems to meet his own interests.

In addition, the simulator may he employed to illustrate problems encountered in kinetic simulation. For example, certain assigned values for rate constants in particular schemes can lead to the requirement for excessive computer time to achieve accurate simulations and in extreme cases to unavoidable errors due to rounding (a phenomenon referred to as stiffness instability). While this may he viewed as an

Figure 5. Autocatalytic conversion of trypsinogen into trypsin. The ordinate is a measure of trypsin concentration in terms of activily units, and the data are those of Kunitzand Northrup ( 1 1 ) . Aoand bare 0.072 units and 0.0003 units. respectively. Values of the rate constants are as indicated in the figure. Simu- lation time was 15 s.

710 1 Joumal of Chemical Education

inherent limitation in the microcomputer-based system, i t PCM84275. We also thank Dr. Roger Millikan for examining should he noted that it is a general problem and other algo- the manuscript and providing helpful comments on the rithms (inchdine the familiar Runre-Kutta method) suffer naoer. . . similar but perhaps less restrictiveconstraints. ~ndeLd, rhis onlv serves to emvhasize the value of dcvelor)inc laree com- Literature Cited . " " puter systems to solve complex reaction schemes. In this sense, (1) Garlinkel,D.,Garfinkel,L..P?inp, M. Glpen.S. and Chance ,~ . ,~nn . ~ e . ~ i o c h a m .

39.473 (1970J. the simulator does not discourage the use of other algorithms ,,, ,, ,,,, ,,, . ,,, ,,,, ,,,, (,,,,,, or larger computers but actually develops a greater appre- (3, woa, ~ . , d w i i l i e m s , ~ . ~ . , ~ . CHEM. E D U C . , ~ I , ~ I ~ ( I ~ ~ ~ ) .

ciation for a variety of simulation techniques. Here again, i t jli ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ , " , " " , " , ~ ~ , " , ~ ; , " , " ; ~ ~ 7 ~ ~ J . is a continuing responsibility of the instructor to point out the (6) ~ d e l s ~ n , D.. computers in chemistry, I. 29 (1976).

merits of all approaches. (7) Shindell, D..and Mapapn~,C.,R~r~r~rdiigsoitheDigitalEplupt Computer uws Socirfy, Vol. 3.2.706 (1976).

(8) Frost, A. A , and Pearson, R. G.. '"Kineti- and Mechanism," J. Wilsy and Sonn,New Acknowledgment YWL, 1965.~. 11.

(9) Alberty, R. A.. in"The Enzymes," (Editors: Myrbaeh, K. Lardy, H., and ~ o y e r . P. D J Development of the simulator was supported in part hy Vol. I , 2nd ~ d . , ~csdemie P ~ S , N ~ W york, 1960, p. 6.

National Institutes of Health Biomedical science support (10J Fromm, H. J., "lnitisl Rate Enzyme Kinetics: Springer~verlsg. Berlin. 1975, pp. 1-5.

Grant RR07099 and National Science Foundation Grant (11J Kunitc, M.,snd ~ o r t h r u p , J. H:, J Gen. ~hysiol., 19.991 (1936).

Volume 55, Number 11, November 1978 / 711