II. Mathematical simulation of ctytochrome oscillations

BIOCHIMICA ET BIOPHYSICAACTA

401

BBA 26485 II. MATHEMATICAL SIMULATION OF CYTOCHROME OSCILLATIONS P. P. GRAYa ANn P. L. ROGERSb aSchool of Chemical Engineering, bSchool of Biological Technology, The University of New .South Wales, P.O. Box ~, Kensington, N.S.W. (Australia)

(Received August iSth, 197o)

SUMMARY The A group cytochrome response patterns following addition and removal of a step input of chloramphenicol to a continuous culture of Candida utilis have been analysed. From the decrease in cytochrome content on chloramphenicol addition, rate constants for cytochrome breakdown (assumed to be unaffected by chloramphenicol) were calculated and shown to depend on the growth rate (viz. k = 0.II h -1 at D = o.I h -1, and k = 0.30 h -1 at D = 0.2 h-l). Following drug removal, the rate of synthesis of A group cytochromes followed a pattern of damped oscillations. This was related to the m R N A concentration which showed a phase lead with respect to the experimentally observed oscillations. Two models proposed by Goodwin were applied to the continuous culture system: (i) cytochrome feedback repression model; (2) product feedback repression model. The fact that oscillations were only generated by Model 2 indicates that a metabolic product causing repression of m R N A synthesis is necessary to explain the experimentally observed response. A decrease in period of oscillation with increase in growth rate was also simulated on the analogue computer.

INTRODUCTION The results presented in the previous paper 1 have shown that it is possible to induce oscillations in the A, B and C group cytochromes of yeast cells by the addition and removal of chloramphenicol. For the A group cytochromes of Candida utilis these oscillations were statistically significant when compared to the variations which occur in steady state continuous culture. In this paper the rates of A group cytochrome breakdown and synthesis are computed from the previously reported data, and the basic feedback loop equations developed by GOODWlN2,3 are tested to see if they explain the observed dynamic response. EXPERIMENTALMETHODS Experimental procedure as outlined in the previous paper was followed. Chloramphenicol concentrations were estimated spectrophotometrically at 275 nm on a Hitachi Perkin-Elmer Model 139. A standard solution of 20 rag/1 chlorBioehim. Biophys. Acta, 230 (I97I) 4Ol-41o

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402

amphenicol gave an absorbance of o.52 with distilled water as reference. The supernatant from the culture was diluted (i in 80 for 2.0 g/1 chloramphenicol) and read against a blank. A calibration curve allowed the concentration of chloramphenicol to be determined. An IBM 162o computer was used in the curve-fitting calculations, and a PACE TR 3 used for the analogue computer simulations. RESULTS AND DISCUSSION

Rate constants for cvtochrome breakdown Experiments with C. utilis have shown that when a step input of chlorampheni-

col was added to the culture, the A and B groups decayed exponentially to their new steady state levels in the presence of the drug 1. However, the time constants for the decrease in concentration were smaller than would have been the case had the excess cytochrome been merely washing out, indicating that active breakdown was occurring. At a dilution rate of 0.2 h -1 for example, the A group cytochromes decayed exponentially with a time constant of 3 h (i.e. time for 63.2% of total change). For direct washout at the same dilution rate, the time constant would be (I/dilution rate) i.e. 5 h. The rate of cytochrome breakdown was determined from the curves b y carrying out a mass balance on the A group cytochromes in the culture vessel; viz. dY dt

-- / ( t ) - - D Y - - k Y

(i)

where ( d Y / d t ) = rate of change of cytochrome concentration (%/unit mass per unit time) ; f (t) = rate of cytochrome synthesis (%/unit mass per unit time) ; Y ~ cytochrome concentration (%/unit mass); D ~ dilution rate (time-I); k ~ rate of breakdown (time-i). This expression assumes that the breakdown rate is first order and is unaffected by chloramphenicol addition. The expression ( d Y / d t ) is a measure of the change in concentration of cytochrome per unit mass of culture per unit time and hence refers to the culture taken as a whole. At the initial steady state before drug addition ( d Y / d t ) = o /:(t) :(D

k)gl

(2)

Subscript I refers to values before drug addition. Similarly for the steady state value in the presence of the drug: /2(t) = (I) + k)Y2

(3)

Subscript 2 refers to values after drug addition. Letting ( d Y / d t ) -- o does not discount the possibility of the concentration of cytochrome in each cell oscillating continuously, but because of the phase differences between individual cells, the concentration of cytochrome in the culture as a whole would appear constant as was observed experimentally. When chloramphenicol was added, the drug would bind to the ribosomes causing a step decrease in the rate of cytochrome synthesis (assuming time lags are small). Unfortunately, however, the drug could not be added as a perfect step and this was taken into account when evaluating the decay constants. The rate of cytoBiochim. Biophys. Acta, 230 (1971) 4Ol-41o

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CYTOCHROME OSCILLATIONS.

403

chrome synthesis would not drop sharply to its steady state value but would mirror the shape of the step input of chloramphenicol. fl(t)x(O÷kJY1 (gl" Y2)(O + k ) i . . . .

f2(t) =to÷ k)Y2

In Table I, the chloramphenicol concentrations taken at different times after the step input are shown. TABLEI D

=

o.zo h -1

T i m e (h)

D

Chloramphenieol eoncn. (g/l)

0.20 h -1

=

T i m e (h)

Chloramphenicol cohen. (g/l)

0

0.00

0

0.00

3.0

1.14

1.54

5"5

1.22

8.5 21.5 45.0

1.5o 1.58 i .60

i.o 3.0 6.5 22.0

1.85

1.9I 1.9o

A curve of best fit using a P O L F I T programme was obtained and allowed the chloramphenicol concentration y(t) to be calculated. The following expression was then written for the rate of change of cytochrome concentration during the transient: dY dt

y ( o o ) - - y(t) -- (D + h ) Y 2 +

(YI--

y(oo)

Y2)(D

+ k) - - ( D

+ k)Y

(4)

where y(t)= chloramphenicol concentration (g/l) at any time, t; y(oo) ----final chloramphenicol concentration (g/l). It can be seen from this expression that as y(t) changes from y(o) to y(oo), the rate of cytochrome synthesis decreases from Eqns. 2 to 3. Eqn. 4 was then used to calculate the breakdown constants (k). Values of dY/dt) were gained by fitting a fifth order polynomial to the A group cytochrome decay curves (Y; Fig. I) and differentiating the resultant expression. The rate constants for A group cytochrome breakdown were evaluated at four

E Eo 4 0

L

8

o 30

E

--~"~==,,,~A,~

*,~ .....

O= .2

hq

8

D = 0 1 h-1 I i - -

E~ <

0

1

2 T i m e (h)

30

I

40

Fig. 1. D e c a y of A g r o u p c y t o c h r o m e s f o r a c h e m o s t a t c u l t u r e of C. utilis f o l l o w i n g a step i n p u t of 2.o g/1 c h l o r a m p h e n i c o l fitted w i t h a P O L F I T p r o g r a m m e . I , d i l u t i o n r a t e o.i h 1; Ak, d i l u t i o n r a t e 0.2 h -t.

Biochim. Biophys. Acta, 230 ( I 9 7 I ) 4Ol 41o

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P . P . GRAY, P . L . ROGERS

different points along t h e d e c a y curve a n d the m e a n value taken. A t a dilution r a t e of o . I o h -I, k ~ o . I I h -1 a n d at a dilution r a t e of 0.20 h -~, the value of k was 0.30 h -~ i n d i c a t i n g t h a t active b r e a k d o w n was occurring. The value of k at D = 0.3 h -a could n o t be d e t e r m i n e d d i r e c t l y due to t h e influence of changing glucose concentrations on the d e c a y curve, b u t was e s t i m a t e d b y linear e x t r a p o l a t i o n to be 0.5 ° h ~.

Rate of synthesis of A group cvtochromes during oscillations F r o m Eqn. I and t h e d e c a y c o n s t a n t s c a l c u l a t e d in t h e previous section, the r a t e of A group c y t o c h r o m e synthesis where oscillatory responses were obtained, was d e t e r m i n e d . Curves were fitted to t h e e x p e r i m e n t a l points b y F o u r i e r Analysis a n d a c o m p u t e r p r o g r a m m e used to calculate values of (dY/dt). K n o w i n g k and D, it was t h e n possible to d e t e r m i n e t h e values of f(t) d u r i n g the oscillations. The results are shown in Figs. 2-4.

•

< 4

o

_

_

_

i

_

~o

2o

30

i

4'0

50

6o

2

Time (h)

Fig. 2. A group c y t o c h r o m e c o n t e n t of C. utilis (D = o,i h -1) following w a s h o u t of 2.0 g/1 chloramphenicol fitted by Fourier analysis. Lower curve is the rate of synthesis of A group c y t o c h r o m e determined from Eqn. i &

35-

"E

' 20

o

o

I a

J: 15

a

o

.<

o~ 2 0

lb

2b Time

~

30

(h)

Fig. 3. A g r o u p c y t o c h r o m e c o n t e n t of C. utilis (D = o.2 h -1) following w a s h o u t of 2.0 g/1 chloramphenicol fitted b y Fourier analysis. L o w e r curve is the rate of synthesis of A group cytochrome. Biochim. Biophys. Acta, 230 (I97 I) 4Ol 4~o

405

(;YTOCHROME OSCILLATIONS. II

In the further analysis of the curves off(t) it is important to note that the spectrophotometric determination of A group cytochromes (a + a3) depends on absorbance by heine a at 598 nm. The oscillations recorded in the cytochrome levels of the A group are really oscillations in the heine a prosthetic group. However, since chloramphenicol is known to influence protein synthesis< 5, and YCAS AND DRABKIN6 have shown the interdependence of heine synthesis and protein synthesis during inhibition, it has been assumed that identical oscillations occur in protein synthesis. The value of Y in Eqn. ~ can now be related to the protein component of the feedback control loop proposed by GOODWlN. Furthermore GOODWI~,"assumes that the rate of synthesis of protein is proportional to the concentration of mRNA, so that the values off(t) will be proportional to the amount of mRNA. Figs. 2 and 4 show that the mRNA component of the control loop will also follow a pattern of damped oscillations following washout of chloramphenicol from C. utilis.

./\ ;\ "

30 C

,

.,

/V': ~ 250

,

/°

r ~

20 ~

<

V I

0

i ,

10

[

'

- - - J

10

20 Time (h)

Fig. 4. A g r o u p c y t o c h r o m e c o n t e n t of C. utilis (D : 0. 3 h -1) fol l ow i ng w a s h o u t of 2.0 g/1 chlor. a m p h e n i c o l f i t t e d b y F o u r i e r a n a l y s i s . L o w e r c u r v e is t h e r a t e of s y n t h e s i s of A g r o u p c yt oc hrorne .

The period of oscillation of the mRNA is similar to that of the A group cytochromes but a phase lead for the mRNA is observed. This is to be expected as the mRNA precedes the protein component in the control loop. The phase lead results from the expression dY /(t) = -dF + (D + k)

so that changes in k and D will alter the phase lead of the mRNA. Experimentally it was observed that a phase lead of approx. 2 h at D = o.i h -1 decreased to a third at a dilution rate of o.3 h -1. Analogue computer simulation

In this section an attempt was made to see whether equations of the form proposed by GOODWlN could explain the experimental findings. The oscillatory responses which followed a disturbance indicated that the system was under some form of dynamic control. However, some doubt exists as to whether cytochrome synthesis is Biochim. Biophys. Acta, 230 (i 97 I) 4 o i - 4 I O

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P . P . GRAY, P. L. ROGERS

controlled by glucose (or a metabolite of glucose) 7 or by the adenosine phosphates 8,9. If glucose were controlling, there would be a feed forward control of cytochrome synthesis in which the concentration of glucose adjusts the set point at the DNA level. If, however, as suggested by POLAKIS AND BARTLEY s, and CHAPMAN ANt) BARTLEY9 the latter is the case, then the system would be under feedback control. GOODWIN'Sgeneral model for control of protein synthesis considers the system under feedback control, and this was assumed for the purposes of analysis. Two basic models developed by GOODWIN were applied to the continuous culture system. (I) Cytochrome feedback repression model In this model, Xi represents the concentration of mRNA coding for the protein components of the A group cytochromes and Yi represents the concentration of these protein components. Feedback repression of mRNA (Xi) is effected by cytochrome (Yi). A mass balance for each component in the continuous system gives: R a t e of change dXi dt d Yi dl

R a t e of -- s y n t h e s i s

R a t e of d e c r e a s e d u e --to

ai -- .4i 2_ k i Y i

breakdown and washout

biXi

= ~fXi - - Pi Y~

(5)

(6)

These equations assume that first order expressions can be used to describe the breakdown and washout of mRNA and A group cytochromes. The terms ai, Ai, ki, bi, ~i and/3i are constants at a given growth rate. The values chosen for the various constants were based on those quoted by GOODWINa. i.e. ai = o.72; Ai ~ o.36; ki -- I.O. The term bi was estimated from the experimental results for C. utilis in continuous culture at a growth rate of o.io h -1. At steady state before drug addition, Yi = 38.4% and using the approximation that Xi 0.2" Yi = 7.7/o, o/ substitution into Eqn. 5 gives bi z o.13 h '. The term fli depends on the rate of breakdown and the rate of washout of A group cytochromes and is equal to (D + k), i.e. fli ~- o.2I at a growth rate of o.Io h -1. The value of ei was based on experimental results for C. utilis and substitution in Eqn. 6. Before drug addition, ~i = 1.15 and following chloramphenicol addition when the steady state concentration of Yi was 7.8%, ~i = o.I3 (assuming that the breakdown rate of mRNA remains constant). In the analogue computer simulation of chloramphenicol addition, the step input was simulated by switching a potentiorrmter from position I set at 1.15 to position 2 set at o.13. (see Fig. 5 for analogue circuit). Washout of the drug was simulated by switching back from 2 to I although in the experimental system a perfect step change would not occur. The results of the simulation involving 2 equations are shown in Fig. 6, and although a number of different constants were tried only a simple overshoot was obtained. The oscillations which had been experimentally observed could not be simulated for a system which proposes direct repression of mRNA synthesis by the A group cytochromes. Biochirn. Biophys. Acta, 230 (1971) 4 o l - 4 1 o

42YTOCHROME OSCILLATIONS. II

407

(II) Product feedback repression model A model involving three simultaneous equations 1° was investigated in which the third equation describes the formation of a metabolic product (Zi) which then closes the control loop by acting as a repressor of m R N A synthesis, dXi dt

ai -- m i --

kiZ i

b i z~'i

(7)

dYi -

-

dt

:

~xiXi -- ~i Yi

(8)

dZi dt

-- YiYi -- biZi -- 6i

(9)

The first two equations are similar to Eqn, 5 and 6 with the metabolic product, Zi, closing the control loop. It would be attractive to think of this as adenosine triphospilate formed by oxidative phosphorylation but this is pure conjecture. In Eqn. 9, t h e r a t e of s y n t h e s i s of Z i is p r o p o r t i o n a l t o t h e c o n c e n t r a t i o n • lOV

-lov

Yi. T h e t e r m

~|Z i

÷lOV

-10 VIi Function switch 1

~ 1

Fig. 5. Analogue computer circuit used to simulate chloramphenicol addition and washout. 05

u 0.4 C

8 0.3 o 02

12

I

L

/~

I

I

I

t

I

I

I

I

Step in Step out Time T. (machine time units)

Fig. 6. Analogue computer plot of the effect of a simulated step input of chloramphenicol on the i_protein component (Yi) of the mcdel. Solution is for model involving two equations. Biochim. Biophys. Acta, 230 (1971) 4Ol-41o

408

P. P. GRAY, P . L . ROGERS

is the rate of removal of product from the culture vessel, and ¢i represents the rate of utilization of end product for growth (expressed per unit mass). The value of ¢i will be a function of the growth rate. In the analogue simulation of this model, constants having the same values as those used in solving Eqn. 5 and 6 were used, together with y~ = o.4, dt -- o.I (equal to dilution rate) and ¢i o.I for C. utilis at a dilution rate of o.I h 1. The results for the protein component (Yi) of the control loop are shown in Fig. 7. The 06 05 ~202h- ~

Q4

8 03 o 0.2

o

I

I

I

i

i

i

-~

Time t ( m a c h i n e t i m e units)

Fig. 7- A n a l o g u e c o m p u t e r plot of t h e c o n c e n t r a t i o n of t h e p r o t e i n c o m p o n e n t (Yi) following t h e s i m u l a t e d r e m o v a l cf a s t e p i n p u t of c h l o r a m p h e n i c o l . T h e solution is for a m o d e l i n v o l v i n g t h r e e e q u a t i o n s a n d i l l u s t r a t e s t h e effect of g r o w t h r a t e on t h e period of t h e oscillations.

initial exponential decrease shown in Fig. 6 was again observed (although not plotted) and the system followed a pattern of damped oscillations when the chloramphenicol was removed. The shapes of the curves for the mRNA (Xi) were again the same as those for Yi, but with a mRNA phase lead. The effect of changing the growth rate was also simulated on the analogue circuit. It was assumed that the rate constants for mRNA and protein synthesis were constant. This implies that at higher growth rates, the increased rate of synthesis is due to an increased concentration. For example, the increase in the rate of protein synthesis would be due to an increased concentration of mRNA, each molecule of mRNA still synthesizing at the same fixed rate. Some evidence for a linear increase in total RNA with growth rate, which would support this hypothesis, has been reported previously for ,4. aerogenes n, 12. The other constants for C. utilis at a growth rate of 0.2 h -a were calculated. The rate of mRNA breakdown (hi) was determined from the experimental results, and the rate of removal of protein (Yi) due to dilution and decay was equivalent to (0.3 -- o.2)Yi. From Eqn. 9 both the term diZi and the term ai will increase with growth rate; and values of 0.2 Zi and 0.2 were substituted. Furthermore, the pre drug levels of the A group cytochromes were all fairly similar at o.I, 0.2 and 0.3 h 1 (ref. I). So it must be assumed that the rate of production of end product per unit concentration of cytochrome, 7~, will also increase with growth rate. A value of 7i = 0.8 was substituted into Eqn. 9. In Fig. 7, a qualitative picture of the effect of growth rate on the period of Biochim. Biophys. Acta, 230 (1971) 4Ol-41o

CYTOCHROME OSCILLATIONS.I I

409

oscillation is presented, and it can be seen that as the growth rate increases from o.I to 0.2 h -1 the period of oscillation decreases. This has been found experimentally to be the case. The picture involving the simulation studies remains qualitative, however, in view of the assumptions made and the arbitrary selection of constants. A further interesting result is shown in Fig. 8, where the degree of damping is shown to he

0.7

06

~k=lD R',,\

O.5 E

0.4

r

/X

~

~,_/~ .,j.

<-..,

.v

8 Q3

2 o,2

I

I

I

I

I

L

L

Time ~ (machine time units) Fig. S. Analogue computer plot of the concentration of the protein component (Yi) following the simulated removal of chloramphenicol showing the effect of varying k in the feedback term ai/(Al + klZi) controlling the synthesis of mRNA (Xi),

influenced by the value of ki in Eqn. 7- The greater the value of ki, i.e. the greater the effect of feedback in the model proposed by GOODWIN, the longer the time before the system stabilizes. CONCLUSIONS By the addition of chloramphenicol to a steady state continuous culture of C. utilis, rate constants for A group cytochrome breakdown have been calculated and shown to be a function of growth rate. Cytochrome breakdown must occur during normal cellular metabolism and is assumed to be unaffected by chloramphenicol addition. Following removal of the drug, the rate of synthesis of A group cytochromes was shown to follow a pattern of damped oscillations. This was related to the m R N A concentrations which showed a phase lead with respect to the oscillations in the level of A group cytochromes. Two models proposed by GOODWlNwere investigated for the continuous culture system and it was established that feedback repression of mRNA synthesis by a metabolic product was necessary for oscillations to occur. Computer simulation showed that the model would predict a decrease in the period of oscillations with increase in the growth rate, and also that the degree of damping depended on the extent of feedback repression. Biochim. Biophys. Acta, 23o (1971) 4ol-41o

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P. P. GRAY, P. L. ROGERS

ACKNOWLEDGEMENTS

The authors wish to acknowledge their indebtedness to Mr. T. Newburn for assistance with the Fourier Analysis, to Mr. P. Souter for assistance with the analogue computer simulations and to Mr. H. Blanch for many fruitful discussions. REFERENCES i 2 3 t 5 6 7 8 9 IO II 12

P. (;RAY AND P. L. ROGERS, Biochim. Biophys. Acta, 23 ° (1971) 396C. GooDwlN, Temporal Organization in Cells, A c a d e m i c Press, N e w Y o r k , i 963 . C. GOODWlN, Advan. Enzyme Regulation, 3 (1965) 425 . M. KROON, Biochim. Biophys. Acta, lO8 (1965) 275. Vt'. LINNANE, D. R. BIGGS, MINTA HUANG AND G. D. CLARK-WALKER,in A. K. MILLS, Aspects of Yeast Metabolism, B l a c k w e l l , Oxford, 1967, p. 217. M. YCAS AND M. DRABKIN, J. Biol. Chem., 224 (1957) 921. F. J. Moss, P. A. D. RICKARD, G. A. BEECH AND F. E. BUSH, Biolech. Bioeng., I i (1969) 56t. E. S. POLAKIS AND W. BARTLEY, Biochem. J., 99 (1966) 52. C. CHAPMAN AND W. BARTLEY, Bioehem. J., i i i (1969) 609. B. C. GOODWlN, Nature, 209 (1966) 479D. HERBERT, in Recent Progress in Microbiology, 7th Intern. Congr., Stockholm, I958, A h n q v i s t a n d W i k s e l l , S t o c k h o l m , 1959, p. 247. A. C. R. DEAN AND P. L. ROGERS. Biochim. Biophys. Acta, 148 (1967) 267. P. B. B. A. A.

Biochim. Biophys. Acta, 23o(1971) 4Ol 41o