J. theor. Biol. (197X) 73, 723-738
Is Nuclear Division in Physmm Controlled by a Continuous Limit Cycle Oscillator? J. TYSON-~ AND W. SACHSENMAIEFC Institut fiir Biochemie und Experimentelle Krebsforschung Universitiit Innsbruck, A-6020 Innsbruck, Austria (Received 31 October 1977, and in revisedform 7 March 1978 ) Analysis of plasmodial fusion experiments shows that the timing of mitosis in the myxomycete Physarumpolycephalumis controlled by a discontinuous(extreme relaxation) oscillator. Furthermore, there is no convincing evidence.that mitosis and DNA synthesis are “downstream”
from the control mechanism. 1. Introduction In the plasmodial stage of its life cycle the slime mold Physarum polycephalumundergoes nuclear division without cell division. Plasmodia, a few
centimeters in diameter, contain about 100 million nuclei, which divide in nearly complete synchrony. For this reason Physarum has been a popular organism for study of the control and the mechanics of nuclear division. From a series of observations of the timing of mitosis after treatment with metabolic inhibitors, u.v.irradiation and plasmodial fusion,Sachsenmaierwas led to suggest a relaxation oscillator model of the control of nuclear division and subsequent DNA synthesis (Sachsenmaier, Remy & Plattner-Schobel, 1972). In this model an activator substance accumulates proportional to total cell mass and is inactivated by binding reversibly to an inhibitor, which is present in an amount proportional to the total nuclear DNA content of the plasmodium. After all the inhibitor units have been “ titrated ” by activator, a signal is generated which triggers mitosis, the onset of DNA synthesis and doubling of the number of inhibitor units. The idea is quite analogous to a mechanical weighing device. Imagine sand slowly dropping into a cup which is counterbalanced by a standard weight. As soon as the weight of sand in the cup exceeds the standard weight, the balance tips over, the cup empties and the balance tips back to collect another aliquot of sand. TPresentaddress:Departmentof Biology, Virginia PolytechnicInstitute and State University, Blacksburg,VA 24061U.S.A. 723
0022-5193/78/0821-0723 $02.00/O
@ 1978AcademicPressInc. (London)Ltd.
724
J. TYSON
AND
W. SACHSENMAIER
Under ideal growth conditions, mitosis occurs quite regularly, every 6-12 h depending on the strain. This basic periodicity led Kauffman to suggest that the timing of mitosis is controlled by an intracellular clock: a continuous limit cycle oscillator (Kauffman, 1974; Kauffman & Wille, 1976a,b). According to this idea, two or more biochemicals interact in such a way as to produce a stable oscillation in cytoplasmic concentrations (cf. Fig. 1). Whenever the concentration of one of these substances reaches a critical value, mitosis and another round of DNA synthesis are initiated. There are two features which distinguish Kauffman’s model from Sachsenmaier’s and a number of other discontinuous models (Fantes,Grant, Pritchard, Sudbery & Wheals, 1975) : (1) The periodic phenomenon is described by a continuous limit cycle oscillator as opposed to a discontinuous relaxation oscillator. This distinction is one of degree since many limit cycle oscillators can be changed continuously from nearly sinusoidal oscillations to nearly discontinuous oscillations simply by changing the value of a system parameter. However, Kauffman and Wille (1976a) make a point that the ” mitotic oscillator ” in P&urum is of moderate relaxation character. That is, over a section of the limit cycle (corresponding to mitosis and the early phase of DNA synthesis) the variables change moderately rapidly, as opposed to extremely rapidly, with respect to changes during later phases of the intermitotic cycle. In particular, approach to the limit cycle is not very rapid, but rather trajectories may spiral around the singular point several times on their way aut to the limit cycle.
-. y, Y6 . \Q FIG. 1. The limit cycle hypothesis. Species Y is produced autocatalytically from X. The singularity( 0) is unstablewith respect to small perturbations.Trajectories spiral out to the closed curve, the limit cycle. .By assumption mitosis is initiated each time Y increases above a critical concentration, Y,.
CONTROL
OF
NUCLEAR
DIVISION
IN
PHYSARUM
725
(2) Since nuclear division and the onset of DNA synthesis are essentially discontinuous phenomena, the events themselves do not play a functional role in generating periodicity in Kauffman’s model (Kauffman & Wille, 1976u,b). The timing of these events is assumed to be controlled by a clock which runs independently of the events, just as our wristwatches continue to run whether or not we eat lunch at noon. Although the second hypothesis is quite controversial, there can be no objection to continuous biochemical oscillations as such. They have been observed repeatedly (Nicolis & Portnow, 1973; Hess & Boiteux, 1974). In three systems: glycolysis in yeast, muscle and red blood cells (Hess & Boiteux, 1974), the aggregation signalling system of a cellular slime mold, Dictyostelium discoideum, (Gerisch & Hess, 1974), and the bromate-malonic acid-cerium reaction, (Field, KiSrBs & Noyes, 1972), the molecular mechanism is known in some detail; and from the mechanism can be demonstrated limit cycle oscillations in fair agreement with experimental observations (Golbeter & Nicolis, 1976; Goldbeter & Segel, 1977; Tyson, 1976). In the absence of such detailed knowledge of the molecular identity of the oscillatory variables and of the nature of their interactions, what sort of evidence can be advanced for limit cycle control? As Winfree (1970, 1973) has shown in his fundamental study of the circadian rhythm of pupal emergence in Drosophila, the best phenomenological evidence for continuous oscillations is the demonstration of the existence of phaseless states. That is, there must exist perturbations which leave the system in a state for which phase, with respect to the oscillatory phenomenon, is undefined (Guckenheimer, 1975). For instance, for the two-variable limit cycle illustrated in Fig. 1, phase is undefined at the steady state, or “ singularity “, inside the limit cycle. The geometry of the set of phaseless points can be much more complicated than this; but, if there is a limit cycle, then there must be some phaseless points somewhere. Winfree (1975) has also emphasized that the existence of critical perturbations, which abolish the observed rhythm, though necessary, is not a sufficient condition that the rhythm be described by a limit cycle oscillator. In terms of a limit cycle model of the timing of mitosis and DNA synthesis, what would be a phaseless state? After a critical perturbation (with heat shock, a pulse of metabolic inhibitor, or fusion with another plasmodium), the periodic phenomenon is not observed, or its periodicity is abolished, for a time long compared to the normal period of the event, under conditions which otherwise are exactly normal. In particular, growth as measured by increase of dry mass or protein content must continue at its normal rate soon after the perturbation is complete. Thus, if the hypothetical control system is driven to a phaseless state, the plasmodium should double in mass several
726
J.
TYSON
AND
W.
SACHSENMAIER
times without concomitant nuclear divisions and DNA synthesis, or with temporally disorganized mitosis and gene replication. Is there any evidence for a phaseless state of the division control system in Physarum ? 2. Fusion Experiments:
Phaseless Points
Pieces cut from plasmodia on different mitotic schedules fuse spontaneously on contact (Guttes, Guttes & Rusch, 1959; Rusch, Sachsemnaier, Behrens & Gruter, 1966). It is usually observed that the fused pair adopts a phase about half way between the phases of the two parent cultures. Let $A and & denote the phases of the parent cultures in the mitotic cycle. For instance, (bA = 3 would mean that one plasmodium was 3 h before mitosis at the time of fusion. (Some authors define phase as hours after the marker event. We prefer the opposite convention because, in Physarum fusion experiments, the time of the previous mitosis is not always determined.) Let $AIB denote the phase of the fused pair, and let: 4&B = @(4A7 w, (1) denote the functional relation among the phases. There is a third variable on which $AIB depends, namely the ratio of the sizes of the plasmodial pieces, A and B. We shall always assume that this ratio is 1 :l, which is the usual case experimentally. Theoretically, 4A,B is the asymptotic phase of the fused pair: i.e. as t -+ co, the fused pair goes through mitosis in synchrony with a control plasmodium of this phase. Since mitosis times in plasmodia synchronous at t = 0 become increasingly spread out as time goes on, it is only practical to determine the time of mitosis within 10-15 h after fusion. Therefore, one wants the fusion to occur as rapidly as possible. This can be accomplished by placing one plasmodium directly on top of the other. Starvation for 60 to 90 min accelerates fusion. Winfree (1974) has pointed out a curious fact about the phase compromise function, @(4,&J. It is reasonable to assume that @(4A, +A) = 4A
(24
and (2b) @(tiA, 4,) = w94?9 4A). From these two assumptions Winfree shows that Cpcannot be a continuous function of $4 and &, i.e. that there exist values of 4A and & for which 4A,B is ill-defined. Furthermore, Winfree shows that the geometry of the locus of singularities of Cp may be used to characterize the underlying mechanism. For instance, for a limit cycle oscillator, like the one in Fig. 1,
Is Nuclear Division in Physmm Controlled by a Continuous Limit Cycle Oscillator? J. TYSON-~ AND W. SACHSENMAIEFC Institut fiir Biochemie und Experimentelle Krebsforschung Universitiit Innsbruck, A-6020 Innsbruck, Austria (Received 31 October 1977, and in revisedform 7 March 1978 ) Analysis of plasmodial fusion experiments shows that the timing of mitosis in the myxomycete Physarumpolycephalumis controlled by a discontinuous(extreme relaxation) oscillator. Furthermore, there is no convincing evidence.that mitosis and DNA synthesis are “downstream”
from the control mechanism. 1. Introduction In the plasmodial stage of its life cycle the slime mold Physarum polycephalumundergoes nuclear division without cell division. Plasmodia, a few
centimeters in diameter, contain about 100 million nuclei, which divide in nearly complete synchrony. For this reason Physarum has been a popular organism for study of the control and the mechanics of nuclear division. From a series of observations of the timing of mitosis after treatment with metabolic inhibitors, u.v.irradiation and plasmodial fusion,Sachsenmaierwas led to suggest a relaxation oscillator model of the control of nuclear division and subsequent DNA synthesis (Sachsenmaier, Remy & Plattner-Schobel, 1972). In this model an activator substance accumulates proportional to total cell mass and is inactivated by binding reversibly to an inhibitor, which is present in an amount proportional to the total nuclear DNA content of the plasmodium. After all the inhibitor units have been “ titrated ” by activator, a signal is generated which triggers mitosis, the onset of DNA synthesis and doubling of the number of inhibitor units. The idea is quite analogous to a mechanical weighing device. Imagine sand slowly dropping into a cup which is counterbalanced by a standard weight. As soon as the weight of sand in the cup exceeds the standard weight, the balance tips over, the cup empties and the balance tips back to collect another aliquot of sand. TPresentaddress:Departmentof Biology, Virginia PolytechnicInstitute and State University, Blacksburg,VA 24061U.S.A. 723
0022-5193/78/0821-0723 $02.00/O
@ 1978AcademicPressInc. (London)Ltd.
724
J. TYSON
AND
W. SACHSENMAIER
Under ideal growth conditions, mitosis occurs quite regularly, every 6-12 h depending on the strain. This basic periodicity led Kauffman to suggest that the timing of mitosis is controlled by an intracellular clock: a continuous limit cycle oscillator (Kauffman, 1974; Kauffman & Wille, 1976a,b). According to this idea, two or more biochemicals interact in such a way as to produce a stable oscillation in cytoplasmic concentrations (cf. Fig. 1). Whenever the concentration of one of these substances reaches a critical value, mitosis and another round of DNA synthesis are initiated. There are two features which distinguish Kauffman’s model from Sachsenmaier’s and a number of other discontinuous models (Fantes,Grant, Pritchard, Sudbery & Wheals, 1975) : (1) The periodic phenomenon is described by a continuous limit cycle oscillator as opposed to a discontinuous relaxation oscillator. This distinction is one of degree since many limit cycle oscillators can be changed continuously from nearly sinusoidal oscillations to nearly discontinuous oscillations simply by changing the value of a system parameter. However, Kauffman and Wille (1976a) make a point that the ” mitotic oscillator ” in P&urum is of moderate relaxation character. That is, over a section of the limit cycle (corresponding to mitosis and the early phase of DNA synthesis) the variables change moderately rapidly, as opposed to extremely rapidly, with respect to changes during later phases of the intermitotic cycle. In particular, approach to the limit cycle is not very rapid, but rather trajectories may spiral around the singular point several times on their way aut to the limit cycle.
-. y, Y6 . \Q FIG. 1. The limit cycle hypothesis. Species Y is produced autocatalytically from X. The singularity( 0) is unstablewith respect to small perturbations.Trajectories spiral out to the closed curve, the limit cycle. .By assumption mitosis is initiated each time Y increases above a critical concentration, Y,.
CONTROL
OF
NUCLEAR
DIVISION
IN
PHYSARUM
725
(2) Since nuclear division and the onset of DNA synthesis are essentially discontinuous phenomena, the events themselves do not play a functional role in generating periodicity in Kauffman’s model (Kauffman & Wille, 1976u,b). The timing of these events is assumed to be controlled by a clock which runs independently of the events, just as our wristwatches continue to run whether or not we eat lunch at noon. Although the second hypothesis is quite controversial, there can be no objection to continuous biochemical oscillations as such. They have been observed repeatedly (Nicolis & Portnow, 1973; Hess & Boiteux, 1974). In three systems: glycolysis in yeast, muscle and red blood cells (Hess & Boiteux, 1974), the aggregation signalling system of a cellular slime mold, Dictyostelium discoideum, (Gerisch & Hess, 1974), and the bromate-malonic acid-cerium reaction, (Field, KiSrBs & Noyes, 1972), the molecular mechanism is known in some detail; and from the mechanism can be demonstrated limit cycle oscillations in fair agreement with experimental observations (Golbeter & Nicolis, 1976; Goldbeter & Segel, 1977; Tyson, 1976). In the absence of such detailed knowledge of the molecular identity of the oscillatory variables and of the nature of their interactions, what sort of evidence can be advanced for limit cycle control? As Winfree (1970, 1973) has shown in his fundamental study of the circadian rhythm of pupal emergence in Drosophila, the best phenomenological evidence for continuous oscillations is the demonstration of the existence of phaseless states. That is, there must exist perturbations which leave the system in a state for which phase, with respect to the oscillatory phenomenon, is undefined (Guckenheimer, 1975). For instance, for the two-variable limit cycle illustrated in Fig. 1, phase is undefined at the steady state, or “ singularity “, inside the limit cycle. The geometry of the set of phaseless points can be much more complicated than this; but, if there is a limit cycle, then there must be some phaseless points somewhere. Winfree (1975) has also emphasized that the existence of critical perturbations, which abolish the observed rhythm, though necessary, is not a sufficient condition that the rhythm be described by a limit cycle oscillator. In terms of a limit cycle model of the timing of mitosis and DNA synthesis, what would be a phaseless state? After a critical perturbation (with heat shock, a pulse of metabolic inhibitor, or fusion with another plasmodium), the periodic phenomenon is not observed, or its periodicity is abolished, for a time long compared to the normal period of the event, under conditions which otherwise are exactly normal. In particular, growth as measured by increase of dry mass or protein content must continue at its normal rate soon after the perturbation is complete. Thus, if the hypothetical control system is driven to a phaseless state, the plasmodium should double in mass several
726
J.
TYSON
AND
W.
SACHSENMAIER
times without concomitant nuclear divisions and DNA synthesis, or with temporally disorganized mitosis and gene replication. Is there any evidence for a phaseless state of the division control system in Physarum ? 2. Fusion Experiments:
Phaseless Points
Pieces cut from plasmodia on different mitotic schedules fuse spontaneously on contact (Guttes, Guttes & Rusch, 1959; Rusch, Sachsemnaier, Behrens & Gruter, 1966). It is usually observed that the fused pair adopts a phase about half way between the phases of the two parent cultures. Let $A and & denote the phases of the parent cultures in the mitotic cycle. For instance, (bA = 3 would mean that one plasmodium was 3 h before mitosis at the time of fusion. (Some authors define phase as hours after the marker event. We prefer the opposite convention because, in Physarum fusion experiments, the time of the previous mitosis is not always determined.) Let $AIB denote the phase of the fused pair, and let: 4&B = @(4A7 w, (1) denote the functional relation among the phases. There is a third variable on which $AIB depends, namely the ratio of the sizes of the plasmodial pieces, A and B. We shall always assume that this ratio is 1 :l, which is the usual case experimentally. Theoretically, 4A,B is the asymptotic phase of the fused pair: i.e. as t -+ co, the fused pair goes through mitosis in synchrony with a control plasmodium of this phase. Since mitosis times in plasmodia synchronous at t = 0 become increasingly spread out as time goes on, it is only practical to determine the time of mitosis within 10-15 h after fusion. Therefore, one wants the fusion to occur as rapidly as possible. This can be accomplished by placing one plasmodium directly on top of the other. Starvation for 60 to 90 min accelerates fusion. Winfree (1974) has pointed out a curious fact about the phase compromise function, @(4,&J. It is reasonable to assume that @(4A, +A) = 4A
(24
and (2b) @(tiA, 4,) = w94?9 4A). From these two assumptions Winfree shows that Cpcannot be a continuous function of $4 and &, i.e. that there exist values of 4A and & for which 4A,B is ill-defined. Furthermore, Winfree shows that the geometry of the locus of singularities of Cp may be used to characterize the underlying mechanism. For instance, for a limit cycle oscillator, like the one in Fig. 1,
