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Departamento de Biologia, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Ribeirão Preto 14040-901, São Paulo, Brazil
* To whom correspondence should be addressed, at Departamento de Biologia, FFCLRP, USP, Avenida Bandeirantes 3900, Ribeirão Preto 14040-901 SP, Brazil. E-mail: mcnamara{at}ffclrp.usp.br
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Abbreviations: A23187, calcium ionophore • BDM, butanedione-monoxime • ER, endoplasmic reticulum • RPCH, red pigment concentrating hormone
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Introduction |
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TOPAbstractIntroductionMaterials and MethodsResults and DiscussionLiterature Cited |
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Within the chromatophores of many amphibians and fish, pigment granules move as independent, individual organelles (Rogers et al., 1997; Rodionov et al., 1998): translocation is rapid and usually involves saltatory movement. However, some shrimp and fish chromatophores contain pigment granules that maintain structural continuity: pigment translocation rates are slower, saltatory movement is apparently absent, and the entire pigment mass migrates as a single, functional unit. Such structural continuity may be provided by ramifying membranous organelles such as the smooth endoplasmic reticulum. This type of structural continuity among pigment granules has been demonstrated in the chromatophores of goldfish and swordtail (Taylor, 1992), in the intertidal shrimp Palaemon affinis (McNamara and Taylor, 1987), and in the neotropical freshwater shrimp Macrobrachium olfersi (McNamara and Sesso, 1982).
Ultrastructural investigation of the red chromatophores on the internal organs of M. olfersi and P. affinis reveals a network of endoplasmic reticulum (ER) cisternae that pervades the cytoplasm, enmeshing and establishing intimate contact with the spherical pigment granules (McNamara, 1980; McNamara and Sesso, 1982; McNamara and Ribeiro, 1999). Pigment aggregation and dispersion in these chromatophores occurs as the unified movement of an entire pigment front, without evidence of saltatory or independent granule movement of any kind (McNamara and Ribeiro, 1999). Such experimental and ultrastructural evidence suggests that the smooth ER in these cells may serve a dual purpose: one functional—that is, protein or macromolecule synthesis; the other structural—that is, binding the pigment granules into a single structural unit.
Biological springs are simple machines
Springs form an integral part of many biological systems, from the macroscopic to the subcellular levels. In their review of biological springs and ratchets, Mahadevan and Matsudaira (2000) describe the spasmoneme spring of Vorticella, which contracts in response to calcium signaling and can propel the cell body at speeds of about 8 cm/s. In Limulus sperm, an actin spring powers the extension of the acrosomal process by regulating the super-coiling (contracted spring state) and relaxation (extended spring state) of actin bundles via a mechanism that does not require ATP or additional actin polymerization (Shin et al., 2007). Many biopolymers that alternately stretch and contract can be described as springs. However, Robert Hooke provided a formal, yet simple mathematical definition of a spring in 1678. He stated that the force of a spring is equal to its rigidity multiplied by its degree of deformation (Hooke, 1678), expressed as stress = strain x (the elastic modulus). Not all elastic substances obey Hooke's Law, however, and ironically, objects exhibiting relatively high elasticity, such as rubberbands, may not behave like true springs. Is it possible that the pigment matrix in the red ovarian chromatophores of M. olfersi behaves like a simple biological spring? To qualify, the matrix must exhibit an elastic component. This may be afforded by the smooth ER, intimately associated with the different pigment granule types throughout the chromatophore cytosol (McNamara and Sesso, 1982; McNamara and Ribeiro, 1999). The structural relevance of the smooth ER here concerns its abundant, convoluted membrane. Taylor (1992) has demonstrated pigment granules in goldfish erythrophores arrayed along narrow ER cisternae, apparently in a stretched state. However, neither the degree of stretch nor the pigment transport kinetics was considered, and we know of no attempts to quantify the elastic properties of ER compression. Numerous reports do take note of the apparent elasticity of biological membranes in general, which may be transiently deformed under positive or negative forces: for example, the tri-dimensional, smooth ER network in Palaemon affinis red epidermal chromatophores collapses after colchicine treatment, forming dense aggregates of cisternae and attached pigment granules (McNamara, 1980). Volume regulatory processes during which cells either swell or shrink on exposure to hyposmotic or hyperosmotic solutions are also particularly good illustrations of membrane elasticity (Larsen et al., 2000; Souza et al., 2000). For a recent review of membrane elasticity and other biophysical properties, see Janmey and Kinnunen (2006).
Qualitative model of pigment translocation in the red ovarian chromatophores of Macrobrachium olfersi
The kinetics of pigment translocation in Macrobrachium olfersi (Wiegmann) red ovarian chromatophores in response to the pigment aggregating hormone, red pigment concentrating hormone (RPCH), in a perfused preparation in vitro, can be divided into at least four distinct phases (McNamara and Ribeiro, 1999) as demonstrated in Figure 1: (1) an initial phase of rapid aggregation in which the pigment front moves at a mean peak velocity of 13.6 µm/min; (2) a final, plateau phase of slow aggregation in which translocation velocity shifts abruptly to 1.7 µm/min; (3) an initial phase of rapid dispersion with a mean peak velocity of 5.0 µm/min; (4) and a final, longer slow phase leading to complete dispersion in which translocation velocity also averages 1.7 µm/min. The calcium ionophore A23187 also produces a pigment-translocation profile with kinetics very similar to RPCH-induced migration (McNamara and Ribeiro, 1999) (see Fig. 1).
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During the slow phase of pigment aggregation in M. olfersi chromatophores, the actin-myosin system appears to compress and charge the pigment matrix with tension, which creates the potential energy released during the rapid phase of pigment dispersion. This aggregated state may result in part from equilibrium between the retrograde myosin-produced force and spring-matrix tension. Evidence for actin-myosin involvement comes from our kinetic studies showing that the myosin ATPase inhibitor, butanedione-monoxime (BDM) (see Higuchi and Takemori, 1989), and the actin-depolymerizing alkaloid, cytochalasin-B, completely inhibit the slow phase but do not affect the fast phase of pigment aggregation in these chromatophores (McNamara and Ribeiro, 1999). These same in vitro experiments, using whole M. olfersi ovaries in a specially designed microperfusion chamber, also reveal that the normal kinetics of pigment dispersion does not take place without prior complete aggregation, which suggests that the fast phase of pigment dispersion is dependent on full spring-matrix compression.
During the slow phase of pigment dispersion, the microtubule-kinesin system may stretch and load the matrix with more tension than occurs during aggregation, owing to asymmetry in the degree of pigment-matrix deformation during dispersion compared to aggregation. The final, dispersed state may represent an equilibrium among three forces: an anterograde/centrifugal, kinesin-derived force; a retrograde/centripetal, spring-matrix tension; and a retrograde/centripetal, myosin-generated force. Evidence that the myosin motor remains active in the dispersed pigment state comes from the demonstration of pigment hyper-dispersion, the surprising phenomenon of pigment translocation and accumulation in the cell periphery. This phenomenon is caused by the inhibition of myosin in live M. olfersi red ovarian chromatophores whose pigment is already dispersed as a result of incubation with polyclonal anti-myosin antibodies (Boyle and McNamara, 2006). Hyper-dispersion has been induced in other chromatophore systems (Rodionov et al., 1998; Nilsson, 2000) and may result from inhibition of part of the pigment-aggregation mechanism. Evidence that kinesin stretches the pigment matrix during dispersion derives from our in vitro perfusion experiments showing that the microtubule-depolymerizing alkaloid colchicine can, per se, trigger immediate, fast aggregation in M. olfersi chromatophores (McNamara and Ribeiro, 1999).
That a considerable degree of pigment movement can be generated independently of RPCH binding, or of externally induced changes in intracellular calcium (via A23187), or of any apparent signaling mechanism, suggests that the ATPase activity of the protein motors in the red ovarian chromatophores of M. olfersi may not be tightly regulated. Such a continuous tug-of-war concept has been proposed previously (Gross et al., 2002). However, pigment aggregation and dispersion in M. olfersi ovarian chromatophores apparently must be highly regulated by some essential process, since experimentally, aggregation and dispersion are usually complete: if not the protein motors themselves, then what process might be involved?
Our recent findings suggest that the microtubule system in M. olfersi ovarian chromatophores may be the target of an intracellular signaling cascade initiated by RPCH binding (Boyle, 2005). Indeed, in our spring-matrix model, proposed microtubule disassembly would disrupt the equilibrium of forces purportedly present in the completely dispersed state. This disruption would lead to immediate, rapid pigment aggregation, since the anterograde/centrifugal, kinesin-derived force would no longer offset the combined spring-matrix and actin-myosin derived, retrograde/centripetal forces. Slow phase aggregation could then recharge the pigment matrix with tension during complete aggregation. On removal of RPCH, the matrix tension is again released, and the rapid phase of pigment dispersion begins, after which slow dispersion reloads the spring-matrix with tension yet again during complete dispersion. This integrated model of pigment translocation suggests an elegant interplay: complete dispersion prepares the pigment matrix for subsequent aggregation, and complete aggregation sets the stage for subsequent pigment dispersion. That dispersion takes nearly twice as long as aggregation (McNamara and Ribeiro, 1999) may be partly due to asymmetry of matrix deformation during complete dispersion and aggregation—that is, the mechanochemical protein motors must generate more tension to deform the pigment matrix to a greater extent during dispersion than is required for aggregation.
The purpose of the present analysis is to employ mathematical methods, together with the classical physical laws developed by Robert Hooke and George Stokes, to corroborate our hypothesis that the nonlinear kinetics characteristic of rapid-phase pigment aggregation and dispersion in the red ovarian chromatophores of Macrobrachium olfersi derives from the release of a passive spring force.
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Materials and Methods |
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TOPAbstractIntroductionMaterials and MethodsResults and DiscussionLiterature Cited |
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Results and Discussion |
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TOPAbstractIntroductionMaterials and MethodsResults and DiscussionLiterature Cited |
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Another important matrix characteristic is its elasticity. Indeed, mathematically, the pigment matrix reasonably obeys Hooke's Law during the rapid phases of aggregation and dispersion (see Table 1), and thus can be considered as a spring that is alternately stretched and compressed.
Semiquantitative modeling of the rapid phases of pigment translocation
Force (stress) and deformation (strain) of springs are directly proportional (Hooke, 1678). According to Hooke's Law (F = –ku), where (F) is the force displacing the spring from equilibrium, (k) is the elastic modulus, or characteristic stiffness, of the spring, and (u) describes the amount of spring deformation. The term ku is negative, reflecting that the force generated by spring deformation is a restoring force, equal to but opposing the deforming force. By combining Hooke's Law with Stokes’ Law (F = –6
rµv) (Larmor, 1907), we can construct a simple proportionality for our system, specific for low Reynolds numbers and spherical pigment granules, that relates the measured maximum velocities of the restoring intracellular spring-matrix to its degree of deformation:
 | (1) |
multiplied by the radius (r) of a sphere moving through a viscous fluid, multiplied by the fluid's viscosity (µ). Viscosity in this case is not simply the chromatophore cytosolic viscosity, but an amalgam of parameters expressed as viscosity. Sources of intracellular friction, pigment granule density, surface area, and viscoelastic properties may contribute to the general resistance of pigment matrix movement: thus, (µ) may be considered as an apparent viscosity. Among the viscoelastic properties of cell cytoplasm is the expression of a range of viscosities, depending on the intracellular force applied. For example, small proteins and vesicles would experience less cytoplasmic drag during transport than, for example, macromolecular chromosomes moving during mitosis. Since pigment transport in palaemonid shrimp chromatophores constitutes bulk cytoplasmic movement, our model assumes that the chromatophore cytosol invariably displays its highest viscosity during these transport phases. In our model, the geometric factor (6r) is specific for calculating the drag on spherical pigment granules, although this term can be modified to represent the drag of other geometric forms, such as chromosomes (Scholey and Mogilner, 2003).
In our system of linear pigment translocation, the term (k/6rµ) from equation (1) is assumed to be constant, since neither spring rigidity (k) nor drag (6rµ) is believed to change appreciably during pigment translocation. Therefore, if the pigment mass were to behave as Hooke defined a spring, in each case of deformation, corresponding to complete dispersion and complete aggregation, the term (v(max)/u(total)) from equation (1) should consistently return the same value, and this is reasonably true for the two parameters measured: for RPCH-induced, rapid-phase aggregation we have 13.6/67 = 0.203, and for rapid-phase dispersion 5.0/25 = 0.200; for A23187-induced, rapid-phase aggregation we find 11.6/54 = 0.215, and for rapid-phase dispersion 6.8/33 = 0.206 (see Table 1). Indeed, the ratios are consistent, which may reflect that, in intracellular environments with low Reynolds numbers, there is little relevant inertia resulting from spring-matrix movement. Under circumstances of high Reynolds numbers, spring inertia would tend to increase the error of displacement measurements—specifically, the points of transition between the rapid and the plateau transport phases—much as a compliant restoring spring with significant inertia might oscillate in periodic movement around its equilibrium point.
The units associated with the ratio (v(max)/u(total)) in our system would normally be expressed as per meter-second (m-1s-1). This is not the case with the ratios calculated above, as the deformation here is relative and dimensionless. Thus, since the previous calculations show that the ratio (v(max)/u(total)) returns a constant value, and because we believe that the ratio (k/6rµ) is also a constant, we conclude that the pigment matrix behaves as Hooke defined a spring. However, if we assume an "ideal" two-dimensional chromatophore as being 20 µm wide across the perikaryon, allowing for a 10-µm diameter nucleus, with a cell extension length of 100 µm, and being roughly triangular in form, as defined by McNamara (1980) for a monochromatic shrimp chromatophore, further calculation allows expression of the ratio in meaningful units: RPCH-induced, rapid-phase aggregation 13.6 µm/min ÷ 530 µm2 = 25.7 nm-1 min-1, and for rapid-phase dispersion 5.0 µm/min ÷ 421.5 µm2 = 11.8 nm-1 min-1; for A23187-induced, rapid-phase aggregation 11.6 µm/min ÷ 385 µm2 = 30.1 nm-1 min-1, and for rapid-phase dispersion 6.8 µm/min ÷ 581.5 µm2 = 11.7 nm–1 min–1 (Table 2).
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The validity of using any instantaneous velocity (v(inst)) during rapid-phase transport, together with its associated deformation, was tested by substituting these respective values into equation (1) and comparing the resulting ratios to (v(max)/u(total)). Let f(x) equal the function originally describing rapid-phase transport (degree of pigment aggregation versus time). For example, for RPCH-induced aggregation f(x) = 92.9 – 17.93x + 1.063x2. The ratio (v(inst)) at time (t) divided by deformation (u) at time (t), would have the form f'(t)/{f(t)/f(8)}, where f(8) is the final time point (8 min after beginning perfusion) of rapid-phase transport (the transition point between rapid and plateau phase aggregation) and is necessary to calculate total deformation. If the use of (v(inst)) and deformation at time (t) is a valid approximation of the ratio (k/6rµ), then the function at various time points should return a horizontal line, f(x) = (v(max)/u(total)). However, f'(t)/{f(t)/f(8)} is a hyperbolic function, asymptotic to the abscissa between the time points 0 and 8 minutes, intercepting the ordinate at (v(max)/u(total)). Thus, to consider (
(inst)) and respective deformation at any time point during rapid aggregation, for the purposes of mathematical modeling, equation (1) may be rewritten for time point (t) as:
 | (2) |
rµ). Note that the measured maximum aggregation velocity does not coincide with total deformation, which contrasts with the values from the mathematical models of aggregation where these parameters are coincident. The cytoplasm of chromatophores and other cells possesses another viscoelastic property known as phase angle that describes the time delay between an applied force and the response of the viscoelastic material to that force. Thus, the maximum transport-velocity response takes time to manifest completely after the velocity-producing force, stored as total deformation, is released. In experimental observations on M. olfersi ovarian chromatosomes perfused with RPCH, maximum velocity is attained at some point between 0 and 2 min after pigment aggregation begins (Fig. 1, and McNamara and Ribeiro, 1999). Velocity then steadily declines, in a nearly linear fashion, as would the force of a restoring spring.
Together, the experimental measurements and our mathematical modeling indicate that the pigment matrix behaves like an intracellular spring of low stiffness operating in a high-drag environment (ratio approximately 1:5, or 0.2), consistent with the experimental observations on transport-phase duration. Eight minutes of spring-restoration time (rapid-phase pigment transport) seems lengthy in the human-scale world of high Reynolds numbers (exceptions are spring-powered clocks and watches, for example, whose restoration to equilibrium is on the order of days). Eight minutes is also a lengthy pigment-translocation period compared to pigment-transport durations in the chromatophores of non-crustacean species. Complete chromatophore pigment translocation in many species, most notably in fish and cephalopods, occurs in a matter of seconds or less, and the mechanisms involved are known to be clearly different from those described here (Demski, 1992; Messenger, 2001). Pigment granules in fish chromatophores are mostly independent, freely diffusible organelles whose transport is rapid, not multiphasic, and usually involves saltatory movement. Notable exceptions are the erythrophores of goldfish and swordtail that contain slow-moving pigment granules bound together by the smooth ER (Taylor, 1992) in a structural continuum similar to that described for Palemon affinis and M. olfersi. Thus, it seems likely that spring-matrix pigment systems exist in animal species other than Crustacea.
Opposing intracellular forces
Inherent to the spring-matrix model is the requirement for a changing equilibrium of opposing intracellular forces to both charge the matrix and release its stored energy. Experimental evidence for these opposing forces has been provided by McNamara and Ribeiro (1999), who showed that the myosin-ATPase inhibitor, BDM, induces resting-state, ovarian chromatosomes to super-disperse their pigments to about 110% of their initial dispersed state. This state reveals an inwardly directed, resting tension that can be eliminated by inhibition of the actin-myosin system, providing evidence for an intra-pigment matrix resting tension, partially mediated by myosin, the force of which is vectored in the retrograde/centripetal direction. Evidence for a centrifugally directed force comes from myosin inhibition by anti-myosin antibodies (Boyle and McNamara, 2006, fig. 4A), in which some resting-state chromatophores also respond by hyper-dispersion, disclosing a continual, anterograde/centrifugal force, possibly kinesin-derived, acting on the pigment matrix. Interestingly, the spring-matrix model can describe both centripetal and centrifugal pigment translocation without attributing an active role to cytoplasmic dynein. This mechanochemical motor is the principal pigment aggregator in vertebrate chromatophores (Nilsson and Wallin, 1997) but apparently is unnecessary for pigment aggregation in cephalopods (Demski, 1992) and crustaceans.
Ultrastructural investigations of the chromatophore cytoskeleton
Electron microscopy is advantageous when investigating crustacean chromatophores with dispersed pigments because the optical properties of the granules impede direct observation of the cytoplasm by fluorescence microscopy (Boyle, 2005; Boyle and McNamara, 2006). Conversely, after aggregation, pigment-free areas in the chromatophore extensions can be examined by fluorescence microscopy. A functional appraisal of the state of the microtubule cytoskeleton thus can be acquired using complementary fluorescence and electron microscopic studies.
Electron microscopy reveals microtubules in M. olfersi ovarian chromatophores with dispersed pigments (McNamara and Sesso, 1982; McNamara and Ribeiro, 1999), while fluorescence microscopy shows the microtubule system to be fragmented and depolymerized in chromatophores with aggregated pigments (Boyle, 2005). These findings suggest a structural rearrangement in the microtubule cytoskeleton coincident with pigment aggregation. McNamara and Taylor (1987) compared microtubule densities in transverse profiles of fully aggregated or fully dispersed epidermal chromatophore extensions in Palemon affinis, revealing an increase in the numerical density of the microtubules per unit area of cytoplasm during aggregation (42 ± 6 microtubules/µm2, 60 ± 2 nm center-to-center spacing; cf., dispersed state 18 ± 2 microtubules/µm2, 54 ± 2 nm center-to-center spacing). Although we could not establish microtubule length by fluorescence microscopy (Boyle, 2005), an increase in the number of microtubule fragments resulting from microtubule dicing associated with pigment aggregation might explain these findings, and would corroborate our theory.
The predominant organization of actin in M. olfersi red ovarian chromatophores appears to be in scattered bundles aligned with the extension long axis, parallel to the microtubules (Ribeiro and McNamara, 2001). Such a spatial organization of the cytoskeleton would be ideal for generating opposing forces, assuming the actin bundles to be of uniform polarity. Given these structural characteristics, a spring-matrix model can be conceived that reasonably describes all the previous experimental findings on pigment translocation in M. olfersi red ovarian chromatophores; the characteristics of the model are described below.
Pigment aggregation
The trigger to release the stress residing in the fully dispersed pigment matrix and to initiate rapid-phase aggregation may involve the RPCH-signaled collapse of the microtubule cytoskeleton. This may occur by microtubule depolymerization, possibly through alterations in GTP availability. An interesting alternative to depolymerization is microtubule dicing into short segments through severing activity by the protein katanin (McNally and Vale, 1993). Regardless of the mechanisms governing their polymerization state, lengthy microtubule segments are present in chromatophores with dispersed pigment (McNamara and Sesso, 1982; McNamara and Ribeiro, 1999), whereas free tubulin and minute microtubule fragments are associated with the aggregated pigment state (Boyle, 2005).
Further, the initial, rapid-phase aggregation may reflect release of stored kinetic energy originally produced by the kinesin/microtubule system during the previous pigment dispersion, and released by microtubule disruption. This stored kinetic energy originates from stretching the pigment matrix like a spring, and is corroborated by the lack of effect of the myosin ATPase inhibitor BDM on rapid-phase aggregation, despite a marked effect on the subsequent plateau phase of aggregation velocity (McNamara and Ribeiro, 1999). It is the plateau phase that may require energy in the form of ATP together with an active actin-myosin system to further compress the pigment matrix during final aggregation. Luby and Porter (1980) showed that erythrophore pigments in the squirrelfish, Holocentris ascensionis, aggregate in an ATP-independent manner, whereas dispersion is ATP-dependent. Regardless of the organelles and motors involved, Luby and Porter reasoned that an ATP-dependent process might store kinetic energy during pigment dispersion, which could be reconverted during aggregation.
Pigment dispersion
In ovarian chromatosome preparations, in vitro, RPCH washout releases the pigment matrix from complete aggregation by an unknown mechanism, which may involve a signaled decrease in myosin activity, possibly via reduced intracellular calcium (McNamara and Ribeiro, 2000), and apparently allows the microtubule system to repolymerize. This in turn permits the anterograde/centrifugal kinesin-produced force to reestablish, and complete dispersion then ensues.
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Acknowledgments |
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Footnotes |
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Literature Cited |
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TOPAbstractIntroductionMaterials and MethodsResults and DiscussionLiterature Cited |
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) or by the calcium ionophore A23187 (25 µmol l–1 A23187, •) in red ovarian chromatophore preparations from the freshwater shrimp Macrobrachium olfersi, perfused in vitro with a 5.5 mmol l–1 calcium-containing Na2HCO3/HEPES-buffered physiological saline. Pigment dispersion was induced by perfusion with a similarly buffered, 2 mmol l–1 EDTA-chelated/calcium-free saline. Pigment translocation occurs in distinct kinetic phases: "rapid-phase" (minutes 22–26) and "slow-phase" (minutes 28–38) pigment aggregation; and "rapid-phase" (minutes 86–94) and "slow-phase" (minutes 96–114) pigment dispersion. The rapid-phase kinetics were used to generate the data in 