Center for Interdisciplinary Mathematics
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# Self-Organization in Biological Systems

September 2, 2013
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Self-organization in a group of organisms is characterized by a global macroscopic order arising out of the local microscopic interactions between individuals in the absence of an external stimulus. This phenomenon occurs in nature in the form of schooling fish, flocking birds, herding mammals, and swarming bacteria. The fundamental question in the study of biological systems is how self-organization (such as collective behavior) arises from the dynamics of the discrete interacting components. One common feature is the formation of large scale coherent structures whose size is much greater than each individual component. To better understand self-organization in biological systems, mathematical models amenable to analysis need to be developed. These models seek to capture the physical mechanisms responsible for collective motion. Techniques from many areas of applied mathematics such as partial differential equations, conservation laws, multiscale analysis, and homogenization theory can be employed to study the origins of this fascinating form of collective behavior.

Our work in this area is motivated by recent experimentally observed phenomena in bacterial suspensions such as collective swimming, enhanced diffusion and mixing, a drastic reduction in viscosity, and the ability to produce useful work from the rotation of microscopic gears [2,3,4,5]. Thus far the study of collective motion has centered on bacteria, which have a simple physical structure. The defining characteristics of a bacterium are its body and several flagella, which exert a force, f_p, on the fluid balanced by a drag force on its body, f_d. Thus, a bacterium can be represented as a force dipole where the two forces are equal in magnitude and opposite in orientation, (see Fig. 1 for the two types of bacteria: Pushers and Pullers, see 1).

Figure 1: Left: Pushers like E. coli (outward force dipole). Right: Pullers like the algae Chlamydomonas swim via a breast stroke (inward force dipole).

Recent models based on differential equations reveal the general self-organization mechanisms behind collective swimming. Common to most models is a coupled PDE/ODE system for the suspension (fluid equations and particle evolution equations respectively). Presently, experiments and theory reveal that self-organization in bacterial suspensions is governed by two competing mechanisms: short-range collisions and long-range hydrodynamic interactions, but the mathematical modeling of both these mechanisms and their interactions remains an open question.

### Recent theoretical work on biosuspensions

Early work has involved studying the drastic seven-fold decrease in the effective viscosity for a bacterial suspension observed at Argonne National Laboratory [2] in contrast to the classic phenomenon of the increase of the effective viscosity in a passive suspension of particles that dated back to the celebrated work of A. Einstein.

Mathematics plays a crucial role in understanding this macroscopic phenomenon through microscopic interactions. Asymptotic analysis and numerical studies of PDE/ODE models (using parallel computing on GPUs) captured the decrease phenomenon. [6,7,8,9,10}. Instead of directly solving a system of ODEs, we employed a kinetic approach that allows for the analysis of the evolution of a probability distribution in phase space for particle positions and orientations. Recent works by other groups have used kinetic equations to analyze collective effects in biological systems [11,12,13,21].

Other fascinating phenomenon is the transition from the individual to the collective state in bacterial suspensions. Experiments of our collaborators from Argonne National Lab reveal the rise of large scale structures when concentration reaches a certain critical threshold [3]. Moreover, it was observed that these structures possess striking universality properties: the correlation length in the collective state remains constant even when the speed of bacteria and their concentration increase, despite an increased injection of energy into the system. This phenomenon was analyzed theoretically in our work [15]. The study of the collective state brings mathematical challenges of the reduction of computational complexity since the bacterial systems are typically very large, 10^{10} bact/cm^3 and their positions are known only with certain degree of certainty (random positions). This challenges have been addressed in [19], where a novel computational approach based on the truncation of BBGKY hierarchy was proposed. The onset of the collective bacterial motion and nonlinear dynamics known as bacterial turbulence has been also studied in 13,16,17,20,22].

### Applications and open questions

Though recent work has led to better knowledge of the origins of self-organization, new approaches still need to be developed for a complete understanding of this phenomenon. Potential future applications include the design of biomimetic functional materials which mimic self-organization found in nature for greater efficiency than achievable by individual components. These applications may include nano-swimmers harvesting the energy of solar light or the ability to control the power of bacterial motion could be used for the development of biomechanical energy conversion systems driven by microorganisms. To move the field forward there are many open questions that need to be answered:

• Which physical mechanisms contribute to the onset of collective motion in biological systems?
• Can we derive from first principles a set of effective equations which govern the motion of ensembles of micro swimmers to explain existing experimental data and even make predictions for future experiments?
• Can simple models for bacteria be generalized to more complex organisms such as fish and birds?
• How to distinguish different types of self-organization observed (e.g., long-range, short-range, etc.)?

As the study of self-organization moves forward one must begin to answer these outstanding questions. Once a more complete picture is known the possibility of the aforementioned applications may begin to be realized.

### References

1. S. Gluzman, D. A. Karpeev and L. Berlyand, Effective viscosity of puller-like microswimmers: a renormalization approach Accepted to J. Royal Society Interface (2013).
2. A. Sokolov and I. S. Aranson, Reduction of Viscosity in Suspension of Swimming Bacteria. Physical Review Letters, 103, 148101 (2009).
3. A. Sokolov, and I. S. Aranson, Physical properties of collective motion in suspensions of bacteria. Physical Review Letters, 109, 248109 (2012).
4. H. H. Wensink , J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Löwen, and J. M. Yeomans, Meso-scale turbulence in living fluids. PNAS, 109, 1-6 (2012).
5. A. Sokolov, I. S. Aranson, J. O. Kessler, and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria. Phys. Rev. Letters, 98, 158102 (2007).
6. B. M. Haines, I. S. Aranson, L. Berlyand, D. A. Karpeev, Effective viscosity of dilute bacterial suspensions: a two-dimensional model. Phys. Biol., 5 046003 (2008).
7. B. M. Haines, A. Sokolov, I. S. Aranson, L. Berlyand, D. A. Karpeev, Three-dimensional model for the effective viscosity of bacterial suspensions. Physical Review E, 80 041922 (2009).
8. B. M. Haines, I. S. Aranson, L. Berlyand, D. A. Karpeev, Effective viscosity of bacterial suspensions: A three-dimensional PDE model with stochastic torque. CPAA, (2012).
9. S. D. Ryan, B. M. Haines, L. Berlyand, F. Ziebert, and I. S. Aranson, Viscosity of bacterial suspensions: Hydrodynamic interactions and self-induced noise. Phys. Rev. E: Rapid Comm., 83, 050904(R) (2011).
10. V. Gyrya, K. Lipnikov, I. Aranson and L. Berlyand, Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations J. Math. Biol. (2010).
11. P. Degond, A. Frouvelle, J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles. J. Nonlinear Science, 23, (2013).
12. S. Motsch, E. Tadmor, A new model for self-organized dynamics and its flocking behavior. J. Statistical Physics, 144, (2011).
13. D. Saintillan and M. J. Shelley Instabilities and Pattern Formation in Active Particle Suspensions: Kinetic Theory and Continuum Simulations. Physical Review Letters, 100, 178103 (2008).
14. S. D. Ryan, B. M. Haines, L. Berlyand, and D. A. Karpeev, A Kinetic Model For Semi-Dilute Bacterial Suspensions. Accepted to SIAM MMS, (2013).
15. S. D. Ryan, A. Sokolov, L. Berlyand, and I. S. Aranson, Correlation properties of collective motion in bacterial suspensions. Submitted to New Journal of Physics, (2013).
16. D. Saintillan and M. J. Shelley, Orientational Order and Instabilities in Suspensions of Self-Locomoting Rods. Physical Review Letters, 99, 058102 (2007).
17. G. Subramanian and D. L. Koch, Critical bacterial concentration for the onset of collective swimming. J. Fluid Mech, 632, 359-400 (2009).
18. D. Saintillan and M. J. Shelley, Emergence of coherent structures and large-scale flows in motile suspensions. J. Royal Society Interface, 9:68, 571-585 (2011).
19. L. Berlyand, P.-E. Jabin, and M. Potomkin, In preparation, (2013).
20. B. Ezhilan, M. J. Shelley and D. Saintillan, Instabilities and nonlinear dynamics of concentrated active suspensions. Phys. of Fluids (2013).
21. E. Lushi, R. E. Goldstein and M. J. Shelley, Collective chemotactic dynamics in the presence of self-generated fluid flows. Phys. Rev. E (2012).
22. M. J. Shelley and C. Hohenegger, Stability of active suspensions. Physical Review E (2010).
23. V. Gyrya, I. S. Aranson, L. V. Berlyand and D. Karpeev, A model of hydrodynamic interaction between swimming bacteria. Bulletin of Mathematical Biology (2010).