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Cell motility: an interplay between local and non-local measurement
BioSystems, 22 (1989) 117-- 126
117
Elsevier Scientific Publishers Ireland Ltd.
Cell motility: an interplay between local and non-local measurement Koichiro Matsuno Technological University of Nagaok~ Nagaoka 9~9-54 {JapanJ (Received February 8th, 1988) (Revision received April 25th, 1988) The bending motions of an ATP-activated actin filament and a flagellar axoneme of starfish spermatozoon exhibit a oneto-many correspondence between the displacement of the medium and the force actualized in the medium, demonstrating sharp contrast to the one-to-one correspondence in classical mechanics. Uniqueness of the actualized forces is lacking. Cell motility suggests the absence of a completely specifiable boundary condition that would unambiguously control the dynamics of generating mechanical forces in motile cells. The one-to-many relationship between the displacement of the medium and the force actualized in the medium necessitates a materialistic capacity for making choices on the part of molecules and cells in transforming future possibilities into actualized reality, the latter of which again serves as the source of the further future possibilities.
Keywords: Actins; Capacity of making choices; Cell motility; Flagellum; Measurement.
1. Introduction
there is no fault with quantum mechanics per se.
Cell motility raises an intriguing theoretical problem in biology. Take, for instance, mechanical forces generated in a motile cell. Then, one may ask whether or not the force distribution inside a motile cell is uniquely determined during development. Tiffs question will, however, be totally dismissed if the formalism of quantum mechanics supplemented by completely definite boundary conditions is literally taken. The time development of any observable in the Heisenberg representation is uniquely determined so long as the boundary condition is completely specified. Mechanical forces are certainly quantum-mechanical observables. Accordingly, uniqueness of mechanical forces actualized in a motile cell would be guaranteed if a completely controllable boundary condition is available to the cell. If not, on the other hand, the deterministic development of mechanical forces could not be defended even though
We shah examine the nature of the boundary condition that makes a motile cell as it is, especially whether completely controllable in principle, though controllable boundary conditions have been widely accepted and practiced in physics (Matsuno, 1988). One method of examining uniqueness or the lack of uniqueness of mechanical forces actualized in a motile cell is to see whether the third law of mechanics is uniquely implemented. For the third law is valid whatever boundary conditions may be applied. If the boundary condition to a motile cell is completely controllable, there would be no arbitrariness in implementing the third law. The present examination requires two different kinds of measurement, one for mechanical forces and the other for the third law. Measurement of mechanical forces is just synonymous with their local genesis due to the interaction with the surroundings. And,
0303-2647/89/$03.50 © 1989 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
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measurement of the third law identified as the null-conservation of acting and reacting forces, though non-local, is an integrated consequence of local interaction variations underlying measurement of local mechanical forces. One thus recognizes that the correlation between the local and non-local measurement plays a key role in coping with the problem of whether completely controllable boundary conditions are available to the dynamics in biology, or a motile cell in particular. 2. Local interaction and non-local measurement
Non-local measurement of the null-conservation of acting and reacting forces is certainly an established characteristic of the relationship existing among measurable mechanical forces. The mutual consistency between local measurement of forces and nonlocal measurement of the null-conservation of acting and reacting forces necessitates local interaction changes for the null-conservation (Matsuno, 1985). Local interaction changes underlying local measurement of forces at two separated regions cannot be communicated simultaneously, but they must be so coordinated as to fulfill the null-conservation of forces a posteriori. When two bodies i and j interact with each other, the null-conservation of forces implies that force f,j acting upon i from j is counterbalanced by f~, satisfying f,) + fj, = O. If force f,j acting from j is varied by the amount of Af,s due to changes in interaction of j with bodies other than ~ the corresponding change of force acting upon 3' from i must follow by the amount of Aft, (= - A f , j). But, this takes time because the communication of interaction changes cannot proceed faster than light. The acting force comes to equilibrate with the reacting force, and vice versa. Local measurement and local genesis of forces proceed in a manner that comes to yield non-local measurement of the null-conservation of forces, thus resulting in the process of force equilibration. Uniqueness of
actualized forces in the medium would be established only at the hypothetical limit of letting the communication velocity of interaction changes diverge. Local interaction changes for the null-conservation of forces will become especially prominent when the communication velocity of the null-conservation is small compared to light velocity~ because these interaction changes become externally measurable in this case. In fact, propagation of the force generator along the flagellar axoneme of a starfish spermatozoon during its bending motion indirectly suggests a possibility that local and non-local measurements of forces may serve as an underlying mechanism of moving the force generator (Hiramoto and Baba, 1978). Referring to measurement process of forces, one can hope to get rid of a tricky question of where does the secondary force to move the generator of the primary force originate. In particular, the rotary movement of bacterial flagellar bundles provides evidence that both the generated torque and the force required for running the rotary motor originate in the same protonmotive force and proton flux (Lowe et al., 1987). 3. M e a s u r e m e n t of forces on m o v i n g coordinates
Local measurement of forces is internally conducive to non-local measurement of the null-conservation of forces. In this regard, classical mechanics requiring a definite one-toone relationship between the displacement of the medium and the force acting upon it is very special in that both local and non-local measurements are accomplished simultaneously in a mutally consistent manner at any moment. This would be equivalent to requiring that the local measurement underlying the non-local one must propagate at an unphysical infinite velocity. On the other hand, however, if local measurements propagate at a finite velocity, as they should, it cannot be tenable to assert that there must be a one-to-one relationship
119 between the displacement and the force over the entire medium. There is no material means to ensure a unique global relationship in an instantaneous manner. The relationship would have to become other than that of a one-to-one correspondence. Cell motility provides a good model for examining whether local measurement really propagates at a finite velocity. One method of estimating the propagation velocity of local measurement of forces is to introduce an observer sitting on a moving body. In particular, an observer sitting on a moving body that follows classical mechanics sees that the resultant force acting upon it always vanishes. For once the displacement of the medium is given, classical mechanics lets the force acting upon it be determined uniquely. W h e n local measurement of forces propagates at a finite velocity, the carrying medium has such a characteristic that the process of becoming counterbalanced between acting and reacting forces propagates in it. The process of becoming counterbalanced is recognized by an observer sitting on a moving body as such a process that the resultant force acting upon the body comes to vanish. Still, the after effect of the process spills over into the neighbourhood, causing a similar movement for the null-conservation of acting and reacting forces in the latter. The spillover of interaction changes is just synonymous with the propagation of local measurements. The process of becoming counterbalanced thus successively propagates in the medium. As a further illustration, let us consider the case of a planar bending motion of a thin filament in a fluid, such as we experience with the cell motility associated with the bending motion of a flagellar axoneme. Suppose a given flexible filament is composed of n equally divided small segments each of which is locally linear. Each segment of length /xl is pulled by the two adjacent segments (see Fig. 1). Put K~ as internal tensile force pulling segment i (i = 1, 2 . . . . . n) at the intersection between i - 1 and i, and 01 as the {smaller) bending angle between
segment i - 1 and i. Then, the bending moment of segment i due to internal tensile forces KI and KI+~ around the moving intersection between i - 1 and i, that is measured by an observer sitting on the moving intersection, turns out to be AM,
= K,.~lO i -
K ~ + 1 /xlO, + l
though the bending moment around the same intersection measured by an observer sitting on a stationary coordinate is - K + 1 /Xle,+l. The only assumption used for obtaining the expression of AM~ is that the mass distribution over each segment is longitudinally homogeneous at least locally. There is also a contribution AM r~si~tiv~ to the bending moment of segment i from the resistive force from the surrounding fluid particles. Each of the bending moments AM1 and AM,resi'tive of segment i around its moving end, when viewed from an observer sitting on the moving end, has to vanish because they have already been involved in the motion according to the second law of mechanics. But the manner of vanishing differs between AM, and I
~M.resiative. i
Fluid mechanics tells us that the resistive force acting upon the filament due to the surrounding fluid particles is uniquely determined once the displacement of the fluid medium is given. The counterbalancing between the resistive force and the corresponding force of inertial reaction from the filament is established at any point on the entire interface between the filament and the surrounding fluid. Local measurement and genesis of the resistive force proceeds simultaneously with non-local measurement of the null-conservation of forces in the whole fluid medium. Thus, the bending m o m e n t / X M resi'tiv¢' always vanishes anywhere on the interface. On the other hand, the way the bending m o m e n t A M ~ due to internal tensile force vanishes, depends upon how local measurement and genesis of the internal force proceeds. W h e n local measurement of the internal ten-
120
to
\
Head
_,ki;Oi i-I
Oi,l/
i_ZI[ F"
"1
Fig. 1. A schematic representation of internal tensile force K, and bending angle 0, of local segment i of length Al in a filament.
sile force propagates at a finite velocity, the bending moment AM, turns out to be in the process of coming to vanish. We shall examine the process of how the bending moment comes to vanish. There are in fact many alternatives in vanishing the bending moment AM~. It can be vanished by adjusting either the bending angles e, and 0~+~ or the internal tensile forces K~ and K~. 1 or both. Let us suppose At is the time interval over which changes in the internal tensile force are propagated over the distance Al. Consequently, adjustments Z~K, =
- AMia~//xlO ,
AO, =
- ~tt~,/KAl
ters a~, fl,, )'i and d~ represent how the bending moment AM i vanishes its own value, although one cannot a priori specify these values. Fundamental to the present dynamics is that these adjustments imputed to the process of vanishing of the bending moment at an arbitrary segment subsequently induces a similar process of vanishing of the bending moment at the adjacent segments. 4. Observed results An interplay between local and non-local measurement of forces can be visualized by examining how the process of becoming counterbalanced between acting and reacting forces, or force equilibration, proceeds in the medium. 4.1. A c t i n f i l a m e n t
Aei+ l = A M i d , / K , + I A I
subject to a, + /~, + Yi + d, = 1 are made over the time interval At. Parame-
We measured the propagation velocity of force equilibration associated with the internal tensile forces generated along actin filaments for both cases of being ATP-activated in the presence of myosin and being not activated. The method was to faithfully retrace
121
the observed bending angles of acting filaments in Fig. 2 (Yanagida et al. 1984) by using the filament model of Fig. 1 and by adjusting the time interval At and parameters % /~, y~ and d~. The reading of the bending angles has been done first by finding the skeleton line that runs right in the middle of the observed filament of a finite width with the aid of an image-processing algorithm eliminating high spatial-frequency components such as noises and blurs and then by associating the bending angle of the skeleton line with that of the real filament. The bending angles of the filament corresponding to the time frame in between two successive pictures in Fig. 2 were estimated based upon the approximation such that the angles vary through a cubic spline function of time over such a short time interval. We understand that the bending motion of actin filaments was planar to the extent it is shown in Fig. 2. The search for the values of % {3,,y~ and d, of each segment at every time interval of At was performed through sampiing out of 1000 or more patterns generated randomly.
Identification of the propagation velocity of force equilibration was made by referring to the standard deviation of internal tensile forces over the similar retracings run independently. If the value of the adjustable parameter A//At is set equal to the propagation velocity of local measurement of internal tensile forces, the most faithful retracement will be realized and the standard deviation will be minimized there at least locally. If the parameter A//At differs from the real propagation velocity in its value, retracement would become less faithful even if available. The larger standard deviation would result because of the random nature of search. Force equilibration eliminates those forces that fail to fulfill the condition for the nullconservation of acting and reacting forces a posteriori. Therefore, it eliminates those hypothetical force equilibrations proceeding at velocities either higher or lower than the real propagation velocity thanks to the selfconstraining capacity intrinsic to force equilibration. The standard deviation of internal tensile forces would thus become minimum at least locally where the adjustable
%"
Fig. 2. Sequences of bending actin filaments in the rigor state (a} and in the ATP-activated state in the presence of myosin (b). The contour length of filaments was 10 ~m. The framing rates of successive pictures were 0.15 s for (a) and 0.1 s for (b). (Reproduced, though slightly modified, from Yanagida et al. (1984), with permission).
122
velocity parameter A//At is equal to the real propagation velocity. Denoting by o, the standard deviation of the internal tensile force K, over retracements run independently, we demonstrate in Figs. 3 and 4 the resultant standard deviation o = ~ + ~ + • • • + °2n-1)1~against the adjustable velocity parameter A//At for two cases, one for the rigor or non-ATP-activated state and the other for the ATP-activated state. In both cases, the actin filaments were divided into 15 segments of equal length for measurement. The propagation velocity of force equilibration associated with an actin filament in the rigor state is to propagate at an infi-
nite velocity as with the resistive force due to the surrounding fluid particles. The internal tensile force is shown to asymptotically have a unique relationship with the displacement of the bending angle. On the other hand, local measurement and genesis of the internal tensile force along the ATP-activated filament in the presence of myosin is found to propagate at velocity 9 ___ 4 ~n/s {see Fig. 4). In fact, the present force
(7 3.0
o" 1.0X107!
2.0
10x106
• AA
1.0 "r 1.OxlOs
2'0 h~,~t (J-zm,,,'~ec)
m
i
i
i
100
50
10
(J'a'n,'&ec) Fig. 3. Standard deviation o of internal tensile forces generated along the actin filament in the rigor state of Fig. 2a against the adjustable velocity parameter Al/At of force equilibration, in which the entire filament was divided into 15 segments of equal length and o was measured among those retracings run independently. Tensile forces internal to the actin filament were measured in arbitrary units. Measurement of standard deviation o was made at the 25th time step in units of At from the initial frame of Fig. 2a. The monotonous decrease of o with the increase of Al/At remained invariant irrespective of the choices of At, Al and the time point at which o was measured.
Fig. 4. Standard deviation o of internal tensile forces generated along the actin filament in the ATP-activated state in the presence of myosin of Fig. 2b against the adjustable velocity parameter Al/At of force equilibration, in which the entire filament was divided into 15 segments of equal length and o was measured among those retracings run independently. Tensile forces internal to the actin filament were measured in arbitrary units. Measurement of standard deviation o was made at about 2.7 s after the initial frame of Fig. 2b. We repeated the similar retracings for the filament divided into 13 (and 17) segments of equal length at about 2.7 s after the initial frame and for the same filament divided into 15 segments of equal length at about 3.2 s {and at about 3.7 s) after the initial frame. The concave character of standard deviation o remained invariant irrespective of the choices of A t Al and the time point at which o was measured.
123
equilibration propagating at velocity 9 _+ 4 ~a/s is empirical evidence supporting local measurement that comes to yield non-local measurement of the null-conservation of acting and reacting forces. The internal tensile force now lacks a unique relationship with the displacement of the bending angle. The present results on force equilibration have been made on the recorded picture of an actin filament. The recorded actin filament is quite enlarged compared with a bare actin filament because of the involvement of fluorescence materials. Nevertheless, we assume that as far as the measurement of the bending angles is concerned, the recorded bending movement would faithfully follow the real movement.
•
4.2. Flagellum
"
-ii
~"~*'~**
h
Force equilibration demonstrated in the ATP-activated actin filament in the presence of myosin provides one representative characteristic of cell motility associated with actomyosin systems, although the latter require a sarcomeric structure of their own when they exhibit motility in muscle contraction. Likewise, a similar motility in the form of force equilibration proceeding at a finite velocity can also be expected for dynein-tubulin systems of flagella. Hiramoto and Baba (1978) have recorded the flagellar movement of spermatozoa of starfish Asterias amurensis with a high speed camera with a framing rate 450/s. Their result is reproduced here in Fig. 5. The record was made of a spermatozoon swimming far from the coverslip surface (~300 p~n) in a trough 1 mm in depth of normal seawater. The flagellar movement was found to be planar. We measured the propagation velocity of force equilibration, A//At. The method was to
Fig. 5. Flagellar movement of a starfish spermatozoon swimming about 300 t~m away from eoverslip in a trough 1 mm in depth of normal seawater at 22°C. The time interval between successive pictures from top to bottom is 2.2 ms. (Reproduced, though slightly modified, from Hiramoto and Baba (1978), with permission).
I
•
50 ~.m
124
faithfully retrace the observed bending angles recorded in Fig. 5 by using the filament model of Fig. 1 and by adjusting the time interval At and parameters a~, ~, y~ and d, as in the previous case of actin filament. When we denote by a, the standard deviation of the internal tensile force K~ over retracings run independently, Fig. 6 exhibits the resultant standard deviation a = (~ + + . . . + ~ _ 1)~ against the adjustable propagation velocity, A//At. We found that the propagation velocity of force equilibration proceeding along the flagellum recorded in Fig. 5 is 720 ± 30 ~m/s. This value is roughly
(7 150
100
5(1
"~t CPm~ec) Fig. 6. Standard deviation o of internal tensile forces among those retracings run independently against the adjustable propagation velocity parameter Al/At of force equilibration, in which the entire flagellar axoneme is divided into 25 segments of equal length. Tensile forces internal to the flagellum were measured in arbitrary units. Measurement of standard deviation o was made at 24.2 ms after the top frame in Fig. 5. Identification of the real propagation velocity of force equilibration remains almost indifferent to the choice of the adjustable parameter of length AI.
three times as large as the free propulsion velocity of the flagellum 259 ± 6 ~m/s (Hiramoto and Baba, 1978). Force equilibration propagating at velocity 720 ± 30 ~m/s again provides evidence that local measurement of forces proceeds in a manner consistent with non-local measurement of the null-conservation of acting and reacting forces. The internal tensile force generated along the flagellar axoneme lacks a unique relationship with the displacement of the bending angle. 5. Discussion Motility associated with ATP-activated actin filaments in the presence of myosin indicates that force equilibration as an interplay between local and non-local measurements of the null-conservation of acting and reacting forces propagates at velocity of several ~m/s. Propagating local measurement and genesis of internal tensile forces is in fact confirmed by another direct observation that an actin filament slides on myosin heads at velocity 10 ± 5/~m/s when ATP is hydrolyzed there (Harada et al., 19871. Since the generation site of internal tensile force is on the contacting inferface between an actin filament and a myosin head (Yanagida et al., 1985), the propagating force equilibration certainly provides indirect evidence of the force generator that is moving as generating forces. A similar propagating force generator was also found with the dyneintubulin system of flagellum in which force equilibration propagates at velocity 720 ± 30 ~m/s along the axoneme. Furthermore, propagation of a force generator suggests that the force-generating units are distributed in the medium. There is in fact empirical evidence that each bacterial flagellar motor contains several force-generating units (Block and Berg, 1984) Propagating force equilibration implies that forces actualized in the medium are not in "the state of' but rather in "the process of' null-conserving of acting and reacting forces. No unique one-t~one relationship is available
125
between the displacement of the medium and the force actualized there. For the one-to-one relationship would let the force be uniquely determined in accordance with the null-conservation of acting and reacting forces in the whole medium at any moment, thus letting force equilibration propagate at an unphysical infinite velocity. Failure of the one-to-one relationship yields instead a one-to-many relationship between the displacement of the medium and the force actualized there. Propagating force equilibration tells us that there is no unique force for a given displacement of the medium. Actualization of the force without being accompanied by its unique correspondence with the displacement of the medium shows that the actualized force is in fact one chosen from many possible alternatives. A materialistic capacity for making choices is latent in the process of force equilibration. It should, however, be emphasized that the present materialistic capacity for making choices refers only to the fact that the actualized force at one location lacks uniqueness in relation to the forces actualized in the whole medium at the same moment. When local genesis of forces underlying non-local measurement of the null-conservation of acting and reacting forces propagates at a finite velocity, the forces actualized at two separated regions are freed from the requirement for uniqueness in reference to null
actin filament and a flagellar axoneme of starfish spermatozoon have suggested that mechanical forces generated in a motile cell do not allow the incorporation of a completely controllable boundary condition for their genesis. Measurement of mechanical forces in the absence of controllable boundary conditions is just an instance of measurement in quantum mechanics, in which measurement is understood to be the process of generating mixed quantum states (see, for instance, Conrad, 1983). If a stationary ensemble of mixed quantum states is available by any chance as in quantum statistical mechanics, one may be able to conceive that a controllable boundary condition, such as a periodic one, be applicable to the ensemble. But, the idea of stationary ensemble does not apply to the intrinsically irreversible process of generating mixed quantum states. Measurement as the on-going process of generating mixed quantum states keeps modifying the boundary condition that might once have been regarded controllable. The process of measurement makes its boundary condition uncontrollable and illusive. Even if the dichotomy of a dynamic process and its boundary condition is overwhelming in physics, the process of measurement in quantum mechanics is undermining the very basis of such a dichotomy. What we have observed in the process of generating mechanical forces in a motile cell is that a materialistic capacity for making choices is one characteristic inherent to measurement as the on-going process of generating mixed quantum states.
Acknowledgment The author is indebted to Toshihiro Suzuki for numerical calculations.
References Block, S.M. and Berg, H.C., 1984, Successive incorporation of force-generating units in the bacterial rotary motor. Nature 309, 470-472.
126 Conrad, M., 1983, Adaptability: The Significance of Variability from Molecule to Ecosystem (Plenum, New York) Chap. 2. Harada, Y., Noguchi, A., Kishino, A. and Yanagida, T., 1987, Sliding movement of single actin filaments on one-headed myosin filaments. Nature 326, 805--808. Hiramoto, Y. and Babe, S.A., 1978, A quantitative analysis of flageUar movement in echinoderm spermatozoa. J. Exp. Biol. 76, 85--104. Lowe, G., Meister, M. and Berg, H.C., 1987, Rapid rotation of flageliar bundles in swimming bacteria. Nature 325, 637- 640.
Matsuno, K., 1985, How can quantum mechanics of material evolution be possible?: symmetry and symmetrybreaking in protobiological evolution. BioSystems 17, 179-192. Matsuno, K., 1988, Protobiology: Physical Basis of Biology (CRC Press, Boca Raton, FL), Chap. 1. Yanagida, T., Nakase, M., Nishiyama, K. and Oosawa, F., 1984, Direct observation of single F-actin filaments in the presence of myosin. Nature 307, 58--60. Yanagida, T., Arata, T. and Oosawa, F., 1985, Sliding distance of actin filament induced by a myosin crossbridge during one ATP hydrolysis cycle. Nature 316, 366369.
117
Elsevier Scientific Publishers Ireland Ltd.
Cell motility: an interplay between local and non-local measurement Koichiro Matsuno Technological University of Nagaok~ Nagaoka 9~9-54 {JapanJ (Received February 8th, 1988) (Revision received April 25th, 1988) The bending motions of an ATP-activated actin filament and a flagellar axoneme of starfish spermatozoon exhibit a oneto-many correspondence between the displacement of the medium and the force actualized in the medium, demonstrating sharp contrast to the one-to-one correspondence in classical mechanics. Uniqueness of the actualized forces is lacking. Cell motility suggests the absence of a completely specifiable boundary condition that would unambiguously control the dynamics of generating mechanical forces in motile cells. The one-to-many relationship between the displacement of the medium and the force actualized in the medium necessitates a materialistic capacity for making choices on the part of molecules and cells in transforming future possibilities into actualized reality, the latter of which again serves as the source of the further future possibilities.
Keywords: Actins; Capacity of making choices; Cell motility; Flagellum; Measurement.
1. Introduction
there is no fault with quantum mechanics per se.
Cell motility raises an intriguing theoretical problem in biology. Take, for instance, mechanical forces generated in a motile cell. Then, one may ask whether or not the force distribution inside a motile cell is uniquely determined during development. Tiffs question will, however, be totally dismissed if the formalism of quantum mechanics supplemented by completely definite boundary conditions is literally taken. The time development of any observable in the Heisenberg representation is uniquely determined so long as the boundary condition is completely specified. Mechanical forces are certainly quantum-mechanical observables. Accordingly, uniqueness of mechanical forces actualized in a motile cell would be guaranteed if a completely controllable boundary condition is available to the cell. If not, on the other hand, the deterministic development of mechanical forces could not be defended even though
We shah examine the nature of the boundary condition that makes a motile cell as it is, especially whether completely controllable in principle, though controllable boundary conditions have been widely accepted and practiced in physics (Matsuno, 1988). One method of examining uniqueness or the lack of uniqueness of mechanical forces actualized in a motile cell is to see whether the third law of mechanics is uniquely implemented. For the third law is valid whatever boundary conditions may be applied. If the boundary condition to a motile cell is completely controllable, there would be no arbitrariness in implementing the third law. The present examination requires two different kinds of measurement, one for mechanical forces and the other for the third law. Measurement of mechanical forces is just synonymous with their local genesis due to the interaction with the surroundings. And,
0303-2647/89/$03.50 © 1989 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
118
measurement of the third law identified as the null-conservation of acting and reacting forces, though non-local, is an integrated consequence of local interaction variations underlying measurement of local mechanical forces. One thus recognizes that the correlation between the local and non-local measurement plays a key role in coping with the problem of whether completely controllable boundary conditions are available to the dynamics in biology, or a motile cell in particular. 2. Local interaction and non-local measurement
Non-local measurement of the null-conservation of acting and reacting forces is certainly an established characteristic of the relationship existing among measurable mechanical forces. The mutual consistency between local measurement of forces and nonlocal measurement of the null-conservation of acting and reacting forces necessitates local interaction changes for the null-conservation (Matsuno, 1985). Local interaction changes underlying local measurement of forces at two separated regions cannot be communicated simultaneously, but they must be so coordinated as to fulfill the null-conservation of forces a posteriori. When two bodies i and j interact with each other, the null-conservation of forces implies that force f,j acting upon i from j is counterbalanced by f~, satisfying f,) + fj, = O. If force f,j acting from j is varied by the amount of Af,s due to changes in interaction of j with bodies other than ~ the corresponding change of force acting upon 3' from i must follow by the amount of Aft, (= - A f , j). But, this takes time because the communication of interaction changes cannot proceed faster than light. The acting force comes to equilibrate with the reacting force, and vice versa. Local measurement and local genesis of forces proceed in a manner that comes to yield non-local measurement of the null-conservation of forces, thus resulting in the process of force equilibration. Uniqueness of
actualized forces in the medium would be established only at the hypothetical limit of letting the communication velocity of interaction changes diverge. Local interaction changes for the null-conservation of forces will become especially prominent when the communication velocity of the null-conservation is small compared to light velocity~ because these interaction changes become externally measurable in this case. In fact, propagation of the force generator along the flagellar axoneme of a starfish spermatozoon during its bending motion indirectly suggests a possibility that local and non-local measurements of forces may serve as an underlying mechanism of moving the force generator (Hiramoto and Baba, 1978). Referring to measurement process of forces, one can hope to get rid of a tricky question of where does the secondary force to move the generator of the primary force originate. In particular, the rotary movement of bacterial flagellar bundles provides evidence that both the generated torque and the force required for running the rotary motor originate in the same protonmotive force and proton flux (Lowe et al., 1987). 3. M e a s u r e m e n t of forces on m o v i n g coordinates
Local measurement of forces is internally conducive to non-local measurement of the null-conservation of forces. In this regard, classical mechanics requiring a definite one-toone relationship between the displacement of the medium and the force acting upon it is very special in that both local and non-local measurements are accomplished simultaneously in a mutally consistent manner at any moment. This would be equivalent to requiring that the local measurement underlying the non-local one must propagate at an unphysical infinite velocity. On the other hand, however, if local measurements propagate at a finite velocity, as they should, it cannot be tenable to assert that there must be a one-to-one relationship
119 between the displacement and the force over the entire medium. There is no material means to ensure a unique global relationship in an instantaneous manner. The relationship would have to become other than that of a one-to-one correspondence. Cell motility provides a good model for examining whether local measurement really propagates at a finite velocity. One method of estimating the propagation velocity of local measurement of forces is to introduce an observer sitting on a moving body. In particular, an observer sitting on a moving body that follows classical mechanics sees that the resultant force acting upon it always vanishes. For once the displacement of the medium is given, classical mechanics lets the force acting upon it be determined uniquely. W h e n local measurement of forces propagates at a finite velocity, the carrying medium has such a characteristic that the process of becoming counterbalanced between acting and reacting forces propagates in it. The process of becoming counterbalanced is recognized by an observer sitting on a moving body as such a process that the resultant force acting upon the body comes to vanish. Still, the after effect of the process spills over into the neighbourhood, causing a similar movement for the null-conservation of acting and reacting forces in the latter. The spillover of interaction changes is just synonymous with the propagation of local measurements. The process of becoming counterbalanced thus successively propagates in the medium. As a further illustration, let us consider the case of a planar bending motion of a thin filament in a fluid, such as we experience with the cell motility associated with the bending motion of a flagellar axoneme. Suppose a given flexible filament is composed of n equally divided small segments each of which is locally linear. Each segment of length /xl is pulled by the two adjacent segments (see Fig. 1). Put K~ as internal tensile force pulling segment i (i = 1, 2 . . . . . n) at the intersection between i - 1 and i, and 01 as the {smaller) bending angle between
segment i - 1 and i. Then, the bending moment of segment i due to internal tensile forces KI and KI+~ around the moving intersection between i - 1 and i, that is measured by an observer sitting on the moving intersection, turns out to be AM,
= K,.~lO i -
K ~ + 1 /xlO, + l
though the bending moment around the same intersection measured by an observer sitting on a stationary coordinate is - K + 1 /Xle,+l. The only assumption used for obtaining the expression of AM~ is that the mass distribution over each segment is longitudinally homogeneous at least locally. There is also a contribution AM r~si~tiv~ to the bending moment of segment i from the resistive force from the surrounding fluid particles. Each of the bending moments AM1 and AM,resi'tive of segment i around its moving end, when viewed from an observer sitting on the moving end, has to vanish because they have already been involved in the motion according to the second law of mechanics. But the manner of vanishing differs between AM, and I
~M.resiative. i
Fluid mechanics tells us that the resistive force acting upon the filament due to the surrounding fluid particles is uniquely determined once the displacement of the fluid medium is given. The counterbalancing between the resistive force and the corresponding force of inertial reaction from the filament is established at any point on the entire interface between the filament and the surrounding fluid. Local measurement and genesis of the resistive force proceeds simultaneously with non-local measurement of the null-conservation of forces in the whole fluid medium. Thus, the bending m o m e n t / X M resi'tiv¢' always vanishes anywhere on the interface. On the other hand, the way the bending m o m e n t A M ~ due to internal tensile force vanishes, depends upon how local measurement and genesis of the internal force proceeds. W h e n local measurement of the internal ten-
120
to
\
Head
_,ki;Oi i-I
Oi,l/
i_ZI[ F"
"1
Fig. 1. A schematic representation of internal tensile force K, and bending angle 0, of local segment i of length Al in a filament.
sile force propagates at a finite velocity, the bending moment AM, turns out to be in the process of coming to vanish. We shall examine the process of how the bending moment comes to vanish. There are in fact many alternatives in vanishing the bending moment AM~. It can be vanished by adjusting either the bending angles e, and 0~+~ or the internal tensile forces K~ and K~. 1 or both. Let us suppose At is the time interval over which changes in the internal tensile force are propagated over the distance Al. Consequently, adjustments Z~K, =
- AMia~//xlO ,
AO, =
- ~tt~,/KAl
ters a~, fl,, )'i and d~ represent how the bending moment AM i vanishes its own value, although one cannot a priori specify these values. Fundamental to the present dynamics is that these adjustments imputed to the process of vanishing of the bending moment at an arbitrary segment subsequently induces a similar process of vanishing of the bending moment at the adjacent segments. 4. Observed results An interplay between local and non-local measurement of forces can be visualized by examining how the process of becoming counterbalanced between acting and reacting forces, or force equilibration, proceeds in the medium. 4.1. A c t i n f i l a m e n t
Aei+ l = A M i d , / K , + I A I
subject to a, + /~, + Yi + d, = 1 are made over the time interval At. Parame-
We measured the propagation velocity of force equilibration associated with the internal tensile forces generated along actin filaments for both cases of being ATP-activated in the presence of myosin and being not activated. The method was to faithfully retrace
121
the observed bending angles of acting filaments in Fig. 2 (Yanagida et al. 1984) by using the filament model of Fig. 1 and by adjusting the time interval At and parameters % /~, y~ and d~. The reading of the bending angles has been done first by finding the skeleton line that runs right in the middle of the observed filament of a finite width with the aid of an image-processing algorithm eliminating high spatial-frequency components such as noises and blurs and then by associating the bending angle of the skeleton line with that of the real filament. The bending angles of the filament corresponding to the time frame in between two successive pictures in Fig. 2 were estimated based upon the approximation such that the angles vary through a cubic spline function of time over such a short time interval. We understand that the bending motion of actin filaments was planar to the extent it is shown in Fig. 2. The search for the values of % {3,,y~ and d, of each segment at every time interval of At was performed through sampiing out of 1000 or more patterns generated randomly.
Identification of the propagation velocity of force equilibration was made by referring to the standard deviation of internal tensile forces over the similar retracings run independently. If the value of the adjustable parameter A//At is set equal to the propagation velocity of local measurement of internal tensile forces, the most faithful retracement will be realized and the standard deviation will be minimized there at least locally. If the parameter A//At differs from the real propagation velocity in its value, retracement would become less faithful even if available. The larger standard deviation would result because of the random nature of search. Force equilibration eliminates those forces that fail to fulfill the condition for the nullconservation of acting and reacting forces a posteriori. Therefore, it eliminates those hypothetical force equilibrations proceeding at velocities either higher or lower than the real propagation velocity thanks to the selfconstraining capacity intrinsic to force equilibration. The standard deviation of internal tensile forces would thus become minimum at least locally where the adjustable
%"
Fig. 2. Sequences of bending actin filaments in the rigor state (a} and in the ATP-activated state in the presence of myosin (b). The contour length of filaments was 10 ~m. The framing rates of successive pictures were 0.15 s for (a) and 0.1 s for (b). (Reproduced, though slightly modified, from Yanagida et al. (1984), with permission).
122
velocity parameter A//At is equal to the real propagation velocity. Denoting by o, the standard deviation of the internal tensile force K, over retracements run independently, we demonstrate in Figs. 3 and 4 the resultant standard deviation o = ~ + ~ + • • • + °2n-1)1~against the adjustable velocity parameter A//At for two cases, one for the rigor or non-ATP-activated state and the other for the ATP-activated state. In both cases, the actin filaments were divided into 15 segments of equal length for measurement. The propagation velocity of force equilibration associated with an actin filament in the rigor state is to propagate at an infi-
nite velocity as with the resistive force due to the surrounding fluid particles. The internal tensile force is shown to asymptotically have a unique relationship with the displacement of the bending angle. On the other hand, local measurement and genesis of the internal tensile force along the ATP-activated filament in the presence of myosin is found to propagate at velocity 9 ___ 4 ~n/s {see Fig. 4). In fact, the present force
(7 3.0
o" 1.0X107!
2.0
10x106
• AA
1.0 "r 1.OxlOs
2'0 h~,~t (J-zm,,,'~ec)
m
i
i
i
100
50
10
(J'a'n,'&ec) Fig. 3. Standard deviation o of internal tensile forces generated along the actin filament in the rigor state of Fig. 2a against the adjustable velocity parameter Al/At of force equilibration, in which the entire filament was divided into 15 segments of equal length and o was measured among those retracings run independently. Tensile forces internal to the actin filament were measured in arbitrary units. Measurement of standard deviation o was made at the 25th time step in units of At from the initial frame of Fig. 2a. The monotonous decrease of o with the increase of Al/At remained invariant irrespective of the choices of At, Al and the time point at which o was measured.
Fig. 4. Standard deviation o of internal tensile forces generated along the actin filament in the ATP-activated state in the presence of myosin of Fig. 2b against the adjustable velocity parameter Al/At of force equilibration, in which the entire filament was divided into 15 segments of equal length and o was measured among those retracings run independently. Tensile forces internal to the actin filament were measured in arbitrary units. Measurement of standard deviation o was made at about 2.7 s after the initial frame of Fig. 2b. We repeated the similar retracings for the filament divided into 13 (and 17) segments of equal length at about 2.7 s after the initial frame and for the same filament divided into 15 segments of equal length at about 3.2 s {and at about 3.7 s) after the initial frame. The concave character of standard deviation o remained invariant irrespective of the choices of A t Al and the time point at which o was measured.
123
equilibration propagating at velocity 9 _+ 4 ~a/s is empirical evidence supporting local measurement that comes to yield non-local measurement of the null-conservation of acting and reacting forces. The internal tensile force now lacks a unique relationship with the displacement of the bending angle. The present results on force equilibration have been made on the recorded picture of an actin filament. The recorded actin filament is quite enlarged compared with a bare actin filament because of the involvement of fluorescence materials. Nevertheless, we assume that as far as the measurement of the bending angles is concerned, the recorded bending movement would faithfully follow the real movement.
•
4.2. Flagellum
"
-ii
~"~*'~**
h
Force equilibration demonstrated in the ATP-activated actin filament in the presence of myosin provides one representative characteristic of cell motility associated with actomyosin systems, although the latter require a sarcomeric structure of their own when they exhibit motility in muscle contraction. Likewise, a similar motility in the form of force equilibration proceeding at a finite velocity can also be expected for dynein-tubulin systems of flagella. Hiramoto and Baba (1978) have recorded the flagellar movement of spermatozoa of starfish Asterias amurensis with a high speed camera with a framing rate 450/s. Their result is reproduced here in Fig. 5. The record was made of a spermatozoon swimming far from the coverslip surface (~300 p~n) in a trough 1 mm in depth of normal seawater. The flagellar movement was found to be planar. We measured the propagation velocity of force equilibration, A//At. The method was to
Fig. 5. Flagellar movement of a starfish spermatozoon swimming about 300 t~m away from eoverslip in a trough 1 mm in depth of normal seawater at 22°C. The time interval between successive pictures from top to bottom is 2.2 ms. (Reproduced, though slightly modified, from Hiramoto and Baba (1978), with permission).
I
•
50 ~.m
124
faithfully retrace the observed bending angles recorded in Fig. 5 by using the filament model of Fig. 1 and by adjusting the time interval At and parameters a~, ~, y~ and d, as in the previous case of actin filament. When we denote by a, the standard deviation of the internal tensile force K~ over retracings run independently, Fig. 6 exhibits the resultant standard deviation a = (~ + + . . . + ~ _ 1)~ against the adjustable propagation velocity, A//At. We found that the propagation velocity of force equilibration proceeding along the flagellum recorded in Fig. 5 is 720 ± 30 ~m/s. This value is roughly
(7 150
100
5(1
"~t CPm~ec) Fig. 6. Standard deviation o of internal tensile forces among those retracings run independently against the adjustable propagation velocity parameter Al/At of force equilibration, in which the entire flagellar axoneme is divided into 25 segments of equal length. Tensile forces internal to the flagellum were measured in arbitrary units. Measurement of standard deviation o was made at 24.2 ms after the top frame in Fig. 5. Identification of the real propagation velocity of force equilibration remains almost indifferent to the choice of the adjustable parameter of length AI.
three times as large as the free propulsion velocity of the flagellum 259 ± 6 ~m/s (Hiramoto and Baba, 1978). Force equilibration propagating at velocity 720 ± 30 ~m/s again provides evidence that local measurement of forces proceeds in a manner consistent with non-local measurement of the null-conservation of acting and reacting forces. The internal tensile force generated along the flagellar axoneme lacks a unique relationship with the displacement of the bending angle. 5. Discussion Motility associated with ATP-activated actin filaments in the presence of myosin indicates that force equilibration as an interplay between local and non-local measurements of the null-conservation of acting and reacting forces propagates at velocity of several ~m/s. Propagating local measurement and genesis of internal tensile forces is in fact confirmed by another direct observation that an actin filament slides on myosin heads at velocity 10 ± 5/~m/s when ATP is hydrolyzed there (Harada et al., 19871. Since the generation site of internal tensile force is on the contacting inferface between an actin filament and a myosin head (Yanagida et al., 1985), the propagating force equilibration certainly provides indirect evidence of the force generator that is moving as generating forces. A similar propagating force generator was also found with the dyneintubulin system of flagellum in which force equilibration propagates at velocity 720 ± 30 ~m/s along the axoneme. Furthermore, propagation of a force generator suggests that the force-generating units are distributed in the medium. There is in fact empirical evidence that each bacterial flagellar motor contains several force-generating units (Block and Berg, 1984) Propagating force equilibration implies that forces actualized in the medium are not in "the state of' but rather in "the process of' null-conserving of acting and reacting forces. No unique one-t~one relationship is available
125
between the displacement of the medium and the force actualized there. For the one-to-one relationship would let the force be uniquely determined in accordance with the null-conservation of acting and reacting forces in the whole medium at any moment, thus letting force equilibration propagate at an unphysical infinite velocity. Failure of the one-to-one relationship yields instead a one-to-many relationship between the displacement of the medium and the force actualized there. Propagating force equilibration tells us that there is no unique force for a given displacement of the medium. Actualization of the force without being accompanied by its unique correspondence with the displacement of the medium shows that the actualized force is in fact one chosen from many possible alternatives. A materialistic capacity for making choices is latent in the process of force equilibration. It should, however, be emphasized that the present materialistic capacity for making choices refers only to the fact that the actualized force at one location lacks uniqueness in relation to the forces actualized in the whole medium at the same moment. When local genesis of forces underlying non-local measurement of the null-conservation of acting and reacting forces propagates at a finite velocity, the forces actualized at two separated regions are freed from the requirement for uniqueness in reference to null
actin filament and a flagellar axoneme of starfish spermatozoon have suggested that mechanical forces generated in a motile cell do not allow the incorporation of a completely controllable boundary condition for their genesis. Measurement of mechanical forces in the absence of controllable boundary conditions is just an instance of measurement in quantum mechanics, in which measurement is understood to be the process of generating mixed quantum states (see, for instance, Conrad, 1983). If a stationary ensemble of mixed quantum states is available by any chance as in quantum statistical mechanics, one may be able to conceive that a controllable boundary condition, such as a periodic one, be applicable to the ensemble. But, the idea of stationary ensemble does not apply to the intrinsically irreversible process of generating mixed quantum states. Measurement as the on-going process of generating mixed quantum states keeps modifying the boundary condition that might once have been regarded controllable. The process of measurement makes its boundary condition uncontrollable and illusive. Even if the dichotomy of a dynamic process and its boundary condition is overwhelming in physics, the process of measurement in quantum mechanics is undermining the very basis of such a dichotomy. What we have observed in the process of generating mechanical forces in a motile cell is that a materialistic capacity for making choices is one characteristic inherent to measurement as the on-going process of generating mixed quantum states.
Acknowledgment The author is indebted to Toshihiro Suzuki for numerical calculations.
References Block, S.M. and Berg, H.C., 1984, Successive incorporation of force-generating units in the bacterial rotary motor. Nature 309, 470-472.
126 Conrad, M., 1983, Adaptability: The Significance of Variability from Molecule to Ecosystem (Plenum, New York) Chap. 2. Harada, Y., Noguchi, A., Kishino, A. and Yanagida, T., 1987, Sliding movement of single actin filaments on one-headed myosin filaments. Nature 326, 805--808. Hiramoto, Y. and Babe, S.A., 1978, A quantitative analysis of flageUar movement in echinoderm spermatozoa. J. Exp. Biol. 76, 85--104. Lowe, G., Meister, M. and Berg, H.C., 1987, Rapid rotation of flageliar bundles in swimming bacteria. Nature 325, 637- 640.
Matsuno, K., 1985, How can quantum mechanics of material evolution be possible?: symmetry and symmetrybreaking in protobiological evolution. BioSystems 17, 179-192. Matsuno, K., 1988, Protobiology: Physical Basis of Biology (CRC Press, Boca Raton, FL), Chap. 1. Yanagida, T., Nakase, M., Nishiyama, K. and Oosawa, F., 1984, Direct observation of single F-actin filaments in the presence of myosin. Nature 307, 58--60. Yanagida, T., Arata, T. and Oosawa, F., 1985, Sliding distance of actin filament induced by a myosin crossbridge during one ATP hydrolysis cycle. Nature 316, 366369.