We have studied a spontaneous self-organization dynamics in a closed, dissipative (in terms of guansine 5′-triphosphate energy dissipation), reaction-diffusion system of acentrosomal microtubules (those nucleated and organized in the absence of a microtubule-organizing centre) multitude constituted of straight and curved acentrosomal microtubules, in highly crowded conditions, in vitro. Our data give experimental evidence that cross-diffusion in conjunction with excluded volume is the underlying mechanism on basis of which acentrosomal microtubule multitudes of different morphologies (straight and curved) undergo a spatial-temporal demix. Demix is constituted of a bifurcation process, manifested as a slow isothermal spinodal decomposition, and a dissipative process of transient periodic spatio-temporal pattern formation. While spinodal decomposition is an energy independent process, transient periodic spatio-temporal pattern formation is accompanied by energy dissipative process. Accordingly, we have determined that the critical threshold for slow, isothermal spinodal decomposition is 1.0 ± 0.05 mg/ml of microtubule protein concentration. We also found that periodic spacing of transient periodic spatio-temporal patterns was, in the overall, increasing versus time. For illustration, we found that a periodic spacing of the same pattern was 0.375 ± 0.036 mm, at 36 °C, at 155th min, while it was 0.540 ± 0.041 mm at 31 °C, and at 275th min after microtubule assembly started. The lifetime of transient periodic spatio-temporal patterns spans from half an hour to two hours approximately. The emergence of conditions of macroscopic symmetry breaking (that occur due to cross-diffusion in conjunction with excluded volume) may have more general but critical importance in morphological pattern development in complex, dissipative, but open cellular systems.
Periodic patterns transiently accompany slow isothermal spinodal decomposition in a closed and dissipative acentrosomal microtubules' reaction-diffusion system constituted by multitude of straight and curved microtubules. Indeed, at highly crowded conditions, i.e., above the bifurcation threshold, cross-diffusion in conjunction with excluded volume is the key driving force of these processes. This finding strongly supports the recent predictions that a variety of transient periodic spatio-temporal structures may emerge in reactive-diffusion systems due to cross-diffusion in conjunction with excluded volume where “the spatial distribution of one species may affect the motion of other species.”1 Acentrosomal microtubules play a critical role in cellular signalling, which is accompanied by cellular morphological changes such as those occurring in dendritic spines during memory formation and learning.2–5 The emergent conditions of symmetry breaking may have quite general but critical importance in morphological pattern development and signal processing in complex, dissipative, and open biological systems. These emergent conditions occur due to a cross-diffusion in conjunction with excluded volume and apart from energy dissipation, which enable transient periodic pattern formation in acentrosomal microtubules multitudes, in reaction-diffusion closed system in vitro. This work confirms as revealed in other related studies that microtubules have great non-linear dynamic capacity to form and reform an abundance of different spatio-temporal patterns, which enables them to process and encode biological signals during cell division and tumorigenesis. Furthermore, this study may be helpful in understanding the fundamental nature of the mechanisms of these phenomena.
We express our sincere gratitude to Professor Maxwell Bennett, AO (Professor of Neuroscience, University Chair and Scientific Director, Brain and Mind Research Institute, The University of Sydney) for his genuine interest in this work and continued support.
We thank microscopist Dr. Ian Kaplin (Australian Key Centre for Microscopy and Microanalysis, The University of Sydney) for technical assistance.
A. The basic physico-chemical (thermodynamics) characteristics of acentrosomal microtubules in vitro
1. Experimental microtubule's system in vitro is closed system
2. Microtubule growth and self-organization: Consumption and dissipation of energy, irreversible hydrolysis of guansine 5′-triphosphate (GTP), and dynamic instability
3. Reaction-diffusion system, excluded volume, and local fluctuations
4. Some relevant findings in this work
B. Biological relevance of acentrosolmal microtubules
II. MATERIALS AND METHODS
A. The spatial extent of microtubule self-organized and non-self-organized pattern
B. The critical threshold
C. The morphology of microtubule self-organized and non-self-organized phase at microscopic scale
A. Referential measurements
B. Demix of microtubules self-organized and non-self-organized phases—a qualitative observation of their durability versus mechanical disruption
C. Demix of self-organized and non-self-organized phase as a function of the MTP concentration—the critical MTP concentration threshold
1. Birefringence observations
2. Light scattering observations
D. Demix of mirotubules self-organized and non-self-organized phases versus time and transient periodic patterns formation and disappearance
IV. EMPIRICAL MODEL
A. Spinodal decomposition
B. Transient spatio-temporal periodic pattern formation
A. Microtubule assembly: Straight and curved microtubules, spontaneous self-organization and demix, turning point, critical threshold, and durability
B. Demix as a bifurcation process
C. The possible mechanism(s) implicated in demixing of mirotubules of different morphology and transient periodic patterns formation
1. Long term demix of straight and curved microtubules: Spinodal decomposition
2. Transient periodic patterns formation
3. Cross-diffusion in the system constituted by multitudes of straight and curved microtubules
- Self organized systems
- Pattern formation
- Self assembly
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