Book Chapter
Reference
Temperature effect on the frequency of simulated intracellular calcium oscillations
DEGLI AGOSTI, Robert
DEGLI AGOSTI, Robert. Temperature effect on the frequency of simulated intracellular calcium oscillations. In: Greppin H., Bonzon M., Degli Agosti R. Some physicochemical and
mathematical tools for understanding of living systems . Genève : University of Geneva, 1993. p. 209-220
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http://archive-ouverte.unige.ch/unige:42881
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Genew, 1993.
TEMPERATURE EFFECT ON THE FREQUENCY OF SIMULATED INTRACELLULAR CALCIUM OSCILLATIONS
R. DEGLI AGOSTI1
INTRODUCTION
Sorne complex properties of living systems can be explained by the theory of non- linear dynamics with simple biochemical reactions [7]. These complex properties do not seem to be directly coded by DNA information, but rather in the dynamics of the interrelationships between different epigenetic basic functions such as membrane transport and/ or enzymatic catalysis. Recently, sorne "minimal" models have been published in this field that describe important biological processes such as cellular division [9] or intracellular calcium oscillations [8; see also 3].
In this article, we are interested in the model of Goldbeter et al. [8] that simulates intracellular calcium oscillations. The effect of temperature will be tested in order to show that the power of a model can also be demonstrated by its capacity to describe new situations that were not normally included in its original formulation.
This aspect is also interesting if we consider the known effect of temperature on biological rhythms. Indeed, in circadian rhythms (period of about 24 hours), the frequency is independent of temperature (e.g., [4]). This is not the case for rhythms with higher frequencies (ultradian rhythms). We would like to answer the following question: is there any tendency in the intracellular calcium auto-oscillation model to compensate for the action of temperature? The answer to this question is not obvious if we consider the equations of such a system (see equations (1) and (2) below). In this situation, the computer simulation with numerical resolution is a simple and elegant way to approach this problem.
1laboratoire de Biochimie et Physiologie végétales, Pavillon des Isotopes, 20 Boulevard d'Yvoy, CH - 1211 Genève 4, Switzerland.
210 DEGLI AGOSTI : Temperature effect and calcium oscillations
DESCRIPTION OF THE MODEL
The model studied here is that elaborated and presented by Goldbeter et al. [8].
The system of differentiai equations describing the dynarnics of the intracellular calcium is the following :
(1)
(2) with
(3)
(4)
Z is the concentration of cytoplasrnic calcium and Y that of reticulum. v0 and kZ are the influx and efflux of this cation in and out of the cell, respectively. The influx rate into reticulum is v2, while v3 is the rate of transport from reticulum to cytoplasm.
The term ~Y accounts for the presence of passive diffusion from reticulum to cytoplasm. The term v 1B represents the calcium efflux from the calcium pool that is sensitive to the InsP3 (inositol 1,4,5-triphosphate) activator, where B varies between 0 and 1 and represents the extent of a stimulation by an extracellular stimulus (e.g.
agonist). It is related to the functioning of the membrane complex that synthesizes InsP 3. All concentrations and rates are expressed with respect to the total cell volume.
v2 and v3 are the equations of the transport rates expressed by Hill equations. V m2
and V ma are the maximal transport rates in and out of the reticulum. ~. KR and KA are the affinity constants for the pumping, release and calcium activation of calcium transport for the reticulum; n, rn and p are the cooperativity of these latter processes.
The initial conditions of this system correspond to the known physiological and experimental values and are 0.2 ,uM for cytoplasmic (Z) and 2 ,uM (Y) for reticulum calcium concentrations, respectively. The values of constants have also been chosen in the physiological range (see [8]) and are : v0 = 1 ,uMs-1, k = 10 s-1, ~ = 1 s-1, v1
= 7.3 ,uMs-1, V m2 = 65 ,uMs-1, V m3 = 500 ,uMs-1, ~ = 1 ,uM, KR = 2 ,uM, KA = 0.9 ,uM, rn = n = 2 and p = 4.
With these conditions, the initiation of oscillations and their modulation in frequency are controlled by the factor B. For more details, see Figures 1 to 3 in [8].
With the exception of the cooperativity factors, ali these constants will be modified by temperature.
THE EFFECT OF TEMPERATURE
The real effect of temperature on each term of the differentiai equations ( 1) and (2) has not been studied experimentally and is therefore not known. This effect can thus only be approached from a theoretical point of view.
The enzymatic reaction rates increase with temperature in the physiological range with a Q10 between 1 and 2 [6], Q10 being the ratio of reaction rates with 10 °C of difference.
The dependency of kinetic constants (k) is often described by the Arrhenius law (e.g., [2]):
K
k(T)
=Ae
T . (5)A and K are constants and T is the absolute temperature.
In the case of reactions (enzymatic reactions, membrane transports) that are described by Hill or Michaelis-Menten (if n= 1) equations of the kind:
(6)
where x is the concentration of a chemical species. There are two parameters (V m and ~ n) that are the complex result of ail individual kinetic constants ki characteri- zing the enzymatic or transport reaction. However, as a first approximation, we can use the formulas given by Ainsworth [1]:
K
~(T)
=ATe
T (7)and
(8) In each of the equations (5), (7) and (8) it is necessary to estimate A and K.
The values of the constants in the differentiai equations (1) and (2) are valid at 37 °C. This corresponds to the experimental temperature where oscillations in calcium are observed (e.g., [11]). Thus, we can write :
(9) (10) (11)
212 DEGLI AGOSTI : Temperature effect and calcium oscillations
where Ffv, FTk and Ffr are the temperature functions that follow the Arrhenius law, according to equations (5), (7) and (8), respectively.
Moreover, the Q10 of these functions must be in the physiological range between 1 and 2. We have chosen the values of 2.0 and 1.5 in arder to study the effect of temperature in a relatively small range around 37 ± 5 °C. We define the Q10 between 32 and 42 °C as Q* 10 and the Q10 between 42 and 32 °C as 0*.10• We can calculate the values of A and K in each situation as follows :
1. V m<T) or its temperature function Ffv:
At 37 °C, V m(T) = V m(37 °C) which means, from (10) : K
FTv = A13
1e
137= 1 ,
where Tt = 273.15 °K + t °C. After transformation of (12), we obtain :
1 -
KA=-e137 137
From (7) and the definition of Q* 10, we have:
That finally gives
K
A.J.;2e 742
K"
A132e
'132K =ln( <to 132) '1;;2132
.
~2'1;;2 -132
By inserting (13) and (15) in (7), we obtain:
( 742'132 )( 1 !_)
FTv =_I_ ( Ca*o 132)
742 -732 '137 -r
137 '1;;2
In the range of temperature studied, it is possible to simplify (16) to:
(12)
(13)
(14)
(15)
(16)
(17)
2. k(T) or its temperature function FTr :
In a manner similar to the above transformations, from (5) we obtain exactly:
(18) which is identical to Ffv.
3. Km(T) or its temperature function FTk:
Similar transformations to those used in point 1, but on equation (8) give:
(19) The graph of Ff [equation (18)] for two values of Q* 10 is given in Figure lA. The Arrhenius plot gives two perfect straight lines (Figure lB). Thus, to simulate the effect of temperature between 32 and 42 °C, it is necessary to multiply, in the system of differentiai equations (1) and (2), the kinetic constants (k) and V max by Ffr, and the affinity constants (~ n) by Ffk. Th us:
-
dZ=FTr(v
0+v
1,B -v
2+v
3 +k1Y-kZ)dt
(20)(21)
with :
-v:: z
Vz - m2
(FTkKf +Z")
(22)-v:: yn Z?
v
3 - m3(FTicK; +
yn)(FTkK~ + Z?)
(23)We assume that the dependency of rates and V m toward temperature is the same for each different process. The same hypothesis is made with the affinity constants Km. This very restrictive hypothesis is necessary as there is no experimental description of the temperature effect on the different elementary processes [i.e. on each constant in the equations (20) and (21)]. It was observed that the multiplication by Ffr or Ffk is almost similar to a linear change of the constants in the small range of temperature studied (Figure lA).
214 DEGLI AGOSTI : Temperature effect and calcium oscillations
3
3 2
cr:s
> .._
.8 0
cr:s
-
FT= ekX0~'9616.B0225'(11310.15-11TJ
A
34
0"10=2.0 0"10= 1.5
38
Temperature (0C)42
1.0
.5
0
-.5
Arrhenius plot of FT
B
0"10=1.5 0"10=2.0~:?5x1o- 3 3.20x1o-
33.25x1o-
33.30x1o-
31/T (°K-1)
Figure l.A: plot of FTv [equation (14)] as a function of temperature. FTv is the temperature function that multiplies rates, kinetic constants and maximum rates in equations (20) and (21). At 37 °C, FTv is equal to 1. FTv is plotted with two Q* 10 values (2.0 and 1.5). B: the temperature function FTv follows the Arrhenius law.
Figure 2 is an example simulating calcium oscillations using equations (20) and (21). The temperature was set at 37 °C, i.e. both Ffr and Ffk are equal to 1. The controlling parameter J3 was progressively changed according to the following schedule:
25, 35, 45, 55, 65, 75, 85, 45 and 25%. Its effect on the appearance and frequency modulation of cytoplasmic calcium oscillations can be clearly observed. The computer program of simulation is interactive, so that any change of J3 or in temperature can be done while the simulation is actively running on the screen. At a low leve! of stimulation (or agonist), no oscillations occur. They only appear at a critical
(triggering) value of hormone, B. Increasing B increases the frequency in a non-linear way.
Another critical step is apparent at a higher concentration of agonist where no more oscillations occur. However, in this case, cytoplasmic calcium is higher than in a non-stimulated cell. Such a complex simulated behaviour is really what is observed in experimental situations (see Figure 5 in [11]). This behaviour is completely reversible in the simulated system (Figure 2).
î.5
î.O
.5
0
lntracellular Calcium oscillations at 37°C
Il
1/.---_____j
1/ 1/ 1/ 1/ 1/i/1/
0 îO 20
Time (sec)
li'-'---
1/
30
l i v - I
40
Figure 2. Simulation of the cytoplasmic calcium oscillation in a living cel! as a function of j3 (25, 35, 45, 55, 65, 75, 85, 45 and 25%). This parameter represents the extent of extracellular stimulation by an agonist. The temperature is 37
°c.
The different modifications of j3 are represented as small bars in the lower part of the graph. The numerical simulation is interactive: the modifications of j3 can be done wlzile the simulation is actively running on the screen.
216 DEGLI AGOSTI : Temperature effect and calcium oscillations
Figure 3 is an example simulating the effect of temperature with B = 45% and Q* 10 = 1.5 for Ffr and Ffk. We can observe as a first result that the madel reacts correctly to theoretical modifications of the parameters in equations (20) and (21).
This is not the case with another madel [11) of intracellular calcium oscillation. With that madel, a small change in temperature-dependent constants by an amount described by equations (18) and (19) leads to a rapid exponential divergence in the numerical resolution of the differentiai equations (result not shawn).
lntracellular Calcium oscillations
(~=45%)
î.5
î.O
.5
11
v V
11v V Il v v v Il
\1If If 1/
1/v
Ilv v v \1
0 0 îO 20 30 40
Time (sec)
Figure 3. Simulated effect of temperature on cytoplasmic calcium oscillations. jJ = 45%
and Q* 10 =1.5 on al! temperature functions (see text). Temperature changes from 32 to 34, 37 and 42 °C, and finally back to 32 °C are indicated by small bars on the lower part of the graph.
The temperature effect on the frequency of intracellular calcium oscillations is shawn in Figure 4A and B together with different values of stimulation (B) and Q* 10•
In each case we . observe a coherent behaviour: the frequency increases regularly with temperature. Arrhenius plots show that the almost linear change in constants results in a non-linear behaviour of the whole integrated system (Figure SA and B).
Finally, Table 1 presents the observed Q* 10 of the oscillation frequency after simulation on the Ffv's and FTk's with different Q* w It can be seen that the madel studied does not have any tendency to compensate for the effect of temperature on the frequency of oscillations. lnstead, it suggests that there could be a small amplification of the temperature effect.
ca++ oscillations 0~=2.0 ca++ oscillations 0~=1.5
3 3
A
a---e beta=35%B
a---e beta=35%e---a beta= 45% e---a be ta= 45%
... _...., beta= 55% ... _...., beta= 55%
'fi
1 1 l'
1./ )'
2 2 ,.r
"
/';/
N''N
/
'N/
6 .; 6
>. 1
./us
>. )l',-'t"'
/,....a
0 ;1 0
c , c
(])
/ /
(])_,
::) ::) / y "
,
'CT , 0"
(]) ,{
/
(]) ,ll/
... p
u: .,
LL
;/
,/ ,cs , z'_,18
-
Î y' , ,
/
,e Î-~
,.0,/
,e' us' ,G'..G',.a"'
, ,e'
.rf , 0 '
us' , , e '
, _a . a ' q
111 ..G''
a'
34
38 42
3438 42
Temperature {°C) Temperature (°C)
Figure 4. Effect of temperature on the frequency of oscillation in cytoplasmic calcium.
A: with Q* 10
=
2.0; B: with Q* 10=
1.5.218 DEGLI AGOSTI : Temperature effect and calcium oscillations
It is obvious that the best way to verify this predicted behaviour for a real system is to perform the experiments. However, this result allows us to discard this kind of mechanism as the basis ofthermocompensation of circadian docks (Q10
=
1.0). On the other hand, we have reported the effect of temperature on the ultradian rhythm of shoot movement in bean [5]. We can calculate from these results a Q10 (20 to 300C) slightly higher than 2.0 (2.3 - 2.5). In this case, it is possible that sorne molecular mechanism may have a component of the same kind as the model of intracellular calcium oscillations.
Arrhenius plot (0~= 2.0) Arrhenius plot (0~= 1.5)
Î.O i.O
"'
\A
v--"' a---a beta= 55% be ta= 45%B
a---a v--"' beta= 55% be ta= 45%"" v.
a----o be ta= 35% a----o be ta= 35%\."' '
~ ...Gl '
"'"'
.5
' "' \
\,__ v_ \ '\\
.5 ~ w._~ '
v.'v"
"'
Q ' "« '
"
~ '
\
\ ~'"'
\ ~
( ) Ill
\
( ) ~c \ \ c::
Q) \ Q)
0 ' 0 '
:J Ill
..,
:J~
o- '
\
o- Cl.~ \ Q ~ ' <9 '
- \ -
'0.. \,']:; \
c
'GI6l \ '
' \ q
\
Q. ' <9\ '
\
'l!J 'Q
\
\ <9 '
-.5 ' \ \ -.5 0..
~
\
C!) \
'a \
\
\ C!)
1:~5x1o- 3
3.20x1o-3 3.25x1o-3 3.30x1o-31:~5x1o- 3
3.20x1o-3 3.25x1o-3 3.30x1o-31/T (°K-1) 1/T (°K-1)
Figure 5. Arrhenius plots. A: data in Figure 4A; B: data in Figure 4B.
Table 1. 0 \0 observed by computer simulation of intracellular calcium oscillations.
Different conditions of temperature are examined on FTv (rates, kinetic constants, maximum rates of transport) and on FTk (affinity constants in Hill equations of transport or activation). Model of Goldbeter et al. [8]. See equations (20) and (21) in text.
Q*w FTr Q*-lo FTk Q * 10 observed
(3=35% (3=45% (3=55%
2.0 2.0 3.16a 2.675 2.446
1.5 1.5
2.159 1.768 1.724
2.0 1.5 2.495 2.389 2.222
1.5 2.0 2.17a 2.049 1.807
a Calculated, as the oscillations of intracellular ca2
+
are not present on the who le range of temperature studied (32-42ÜC).CONCLUSION
The availability in the scientific literature of very simple ("minimal") models such as those presented by Goldbeter et al. [8] and Goldbeter [9] should have a very important impact. These models are based on very common biochemical reactions that are widely understood among biologists. The heart of sorne complex behaviours is shown to be the integration in a network or structure of these very elementary biochemical reactions. Such models appear not only to describe in fine detail the behaviour of biological systems but can also be used in new situations with a predictive power (e.g. temperature effect, as in this work, or the effect of electromag- netic fields in [10]). This is now possible with the current availability of powerful persona! computers. In this respect, the apparent simplicity in the use of such models will certainly determine the accessibility and thus the diffusion to researchers, teachers and students of the fundamental aspects of non-linear dynamics in biology and other fields.
220 DEGLI AGOSTI : Temperature effect and calcium oscillations
REFERENCES
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[3] Berridge MJ, Irvine RF (1989) Inositol phosphates and cell signalling. Nature 341: 197-205.
[4] Daan S (1982) Circadian rhythms in animais and plants. In: J Brady , ed, Biological Timekeeping. Cambridge University Press, Cambridge, pp 11-32.
[5] Degli Agosti R, Millet B (1991) Influence of environmental factors on the ultradian rhythm of shoot movement in Phaseolus vulgaris L. J Interdiscipl Cycle Res 22: 325-332.
[6] Dixon M, Webb EC (1964) Enzymes. Longmans, London, p 158.
[7] Goldbeter A (1990) Rythmes et chaos dans les systèmes biochimiques et cellulaires. Masson, Paris.
[8] Goldbeter A, Dupont G, Berridge MJ (1990) Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphoryla- tion. Proc Nat! Acad Sei USA 87: 1461-1465.
[9] Goldbeter A (1991) A minimal cascade mode! for the mitotic oscillator involving cyclin and cdc2 kinase. Proc Natl Acad Sei USA 88: 9107-9111.
[10] Grundler W, Kaiser F, Keilmann F, Walleczek J (1992) Mechanisms of electromagnetic interaction with cellular systems. Naturwissenschaften 79: 551- 559.
(11] Somogyi R, Stucki JW (1991) Hormone-induced calcium oscillations in liver cells can be explained by a simple one pool model. J Biol Chem 266: 11068-11077.
