A Bayesian Framework to Estimate Fluid and Material Parameters in Micro-swimmer Models

This article has been updated

Abstract

To advance our understanding of the movement of elastic microstructures in a viscous fluid, techniques that utilize available data to estimate model parameters are necessary. Here, we describe a Bayesian uncertainty quantification framework that is highly parallelizable, making parameter estimation tractable for complex fluid–structure interaction models. Using noisy in silico data for swimmers, we demonstrate the methodology’s robustness in estimating fluid and elastic swimmer parameters, along with their uncertainties. We identify correlations between model parameters and gain insight into emergent swimming trajectories of a single swimmer or a pair of swimmers. Our proposed framework can handle data with a spatiotemporal resolution representative of experiments, showing that this framework can be used to aid in the development of artificial micro-swimmers for biomedical applications, as well as gain a fundamental understanding of the range of parameters that allow for certain motility patterns.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Change history

  • 10 February 2021

    Funding information was corrected

References

  1. Ahmadi E, Cortez R, Fujioka H (2017) Boundary integral formulation for flows containing an interface between two porous media. J Fluid Mech 816:71–93

    MathSciNet  MATH  Google Scholar 

  2. Auriault JL (2009) On the domain of validity of Brinkman’s equation. Trans Porous Media 79:215–223

    MathSciNet  Google Scholar 

  3. Beck JL, Yuen KV (2004) Model selection using response measurements: Bayesian probabilistic approach. J Eng Mech 130(2):192–203

    Google Scholar 

  4. Bowman C, Larson K, Roitershtein A, Stein D, Matzavinos A (2018) Bayesian uncertainty quantification for particle-based simulation of lipid bilayer membranes. Springer, Cham, pp 77–102

    Google Scholar 

  5. Brinkman HC (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of paticles. Appl Sci Res 1:27–34

    MATH  Google Scholar 

  6. Carichino L, Olson S (2019) Emergent three-dimensional sperm motility: coupling calcium dynamics and preferred curvature in a Kirchhoff rod model. J Math Med Biol 36:439–469

    MathSciNet  Google Scholar 

  7. Ching J, Chen Y (2007) Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J Eng Mech 133:816–832

    Google Scholar 

  8. Cortez R (2001) The method of regularized Stokeslets. SIAM J Sci Comput 23:1204–1225

    MathSciNet  MATH  Google Scholar 

  9. Cortez R, Cummins B, Leiderman K, Varela D (2010) Computation of three-dimensional Brinkman flows using regularized methods. J Comput Phys 229:7609–7624

    MathSciNet  MATH  Google Scholar 

  10. Dasgupta M, Liu B, Fu H, Berhanu M, Breuer K, Powers T, Kudrolli A (2013) Speed of a swimming sheet in Newtonian and viscoelastic fluids. Phys Rev E 87:013015

    Google Scholar 

  11. Dillon R, Fauci L, Yang X (2006) Sperm motility and multiciliary beating: an integrative mechanical model. Comput Math Appl 52:749–758

    MathSciNet  MATH  Google Scholar 

  12. Durlofsky L, Brady JF (1987) Analysis of the Brinkman equation as a model for flow in porous media. Phys Fluids 30(11):3329–3341

    MATH  Google Scholar 

  13. Elfring G, Lauga E (2009) Hydrodynamic phase locking of swimming microorganisms. Phys Rev Lett 103:088101

    Google Scholar 

  14. Elfring G, Lauga E (2011a) Passive hydrodynamic synchronization of two-dimensional swimming cells. Phys Fluids 23:011902

    Google Scholar 

  15. Elfring G, Lauga E (2011b) Synchronization of flexible sheets. J Fluid Mech 674:163–173

    MathSciNet  MATH  Google Scholar 

  16. Elfring G, Pak O, Lauga E (2010) Two-dimensional flagellar synchronization in viscoelastic fluids. J Fluid Mech 646:505–515

    MathSciNet  MATH  Google Scholar 

  17. Elgeti J, Kaupp U, Gompper G (2010) Hydrodynamics of sperm cells near surfaces. Biophys J 99(4):1018–1026

    Google Scholar 

  18. Elgeti J, Winkler R, Gompper G (2015) Physics of microswimmers—single particle motion and collective behavior: a review. Rep Prog Phys 78:056601

    MathSciNet  Google Scholar 

  19. Fauci L, McDonald A (1995) Sperm motility in the presence of boundaries. Bull Math Biol 57:679–699

    MATH  Google Scholar 

  20. Flemming H, Wingender J (2010) The biofilm matrix. Nat Rev Microbiol 8:623–633

    Google Scholar 

  21. Fu H, Powers T, Wolgemuth C (2007) Theory of swimming filaments in viscoelastic media. Phys Rev Lett 99:258101–05

    Google Scholar 

  22. Fu H, Wolgemuth C, Powers T (2009) Swimming speeds of filaments in nonlinearly viscoelastic fluids. Phys Fluids 21:033102

    MATH  Google Scholar 

  23. Fu H, Shenoy V, Powers T (2010) Low Reynolds number swimming in gels. Europhys Lett 91:24002

    Google Scholar 

  24. Gadelha H, Gaffney E, Goriely A (2013) The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles. Proc Natl Acad Sci USA 110:12180–12195

    Google Scholar 

  25. Gaffney EA, Gadêlha H, Smith DJ, Blake JR, Kirkman-Brown JC (2011) Mammalian sperm motility: observation and theory. Annu Rev Fluid Mech 43:501–528

    MathSciNet  MATH  Google Scholar 

  26. Gallagher M, Smith D, Kirkman-Brown J, Cupples G (2020) Fast. https://www.flagellarcapture.com. Accessed 29 Nov 2020

  27. Gao W, Wang J (2014) Synthetic micro/nanomotors in drug delivery. Nanoscale 6:10486–10494

    Google Scholar 

  28. Hadjidoukas P, Angelikopoulos P, Papadimitriou C, Koumoutsakos P (2015) \(\Pi \)4U: a high performance computing framework for Bayesian uncertainty quantification of complex models. J Comput Phys 284:1–21

    MathSciNet  MATH  Google Scholar 

  29. Ho N, Leiderman K, Olson S (2016) Swimming speeds of filaments in viscous fluids with resistance. Phys Rev E 93(4):043108

    Google Scholar 

  30. Ho N, Leiderman K, Olson S (2019) A 3-dimensional model of flagellar swimming in a Brinkman fluid. J Fluid Mech 864:1088–1124

    MathSciNet  MATH  Google Scholar 

  31. Howells ID (1974) Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J Fluid Mech 64:449–475

    MATH  Google Scholar 

  32. Huang J, Carichino L, Olson S (2018) Hydrodynamic interactions of actuated elastic filaments near a planar wall with applications to sperm motility. J Coupled Syst Multiscale Dyn 6:163–175

    Google Scholar 

  33. Ishimoto K, Gaffney E (2018a) An elastohydrodynamical simulation study of filament and spermatozoan swimming driven by internal couples. IMA J Appl Math 83:655–679

    MathSciNet  MATH  Google Scholar 

  34. Ishimoto K, Gaffney E (2018b) Hydrodynamic clustering of human sperm in viscoelastic fluids. Sci Rep 8:15600

    Google Scholar 

  35. Jeznach C, Olson S (2020) Dynamics of swimmers in fluids with resistance. Fluids 5(14):1–20

    Google Scholar 

  36. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems, vol 160. Springer, Berlin

    Google Scholar 

  37. Larson K, Bowman C, Papadimitriou C, Koumoutsakos P, Matzavinos A (2019a) Detection of arterial wall abnormalities via Bayesian model selection. R Soc Open Sci 6:182229

    Google Scholar 

  38. Larson K, Zagkos L, Auley MM, Roberts J, Kavallaris NI, Matzavinos A (2019b) Data-driven selection and parameter estimation for DNA methylation mathematical models. J Theor Biol 467:87–99

    MathSciNet  MATH  Google Scholar 

  39. Lauga E (2007) Propulsion in a viscoelastic fluid. Phys Fluids 19:083104

    MATH  Google Scholar 

  40. Lauga E, Powers T (2009) The hydrodynamics of swimming microorganisms. Rep Prog Phys 72:096601

    MathSciNet  Google Scholar 

  41. Leiderman K, Olson S (2016) Swimming in a two-dimensional Brinkman fluid: computational modeling and regularized solutions. Phys Fluids 28(2):021902

    Google Scholar 

  42. Leiderman K, Olson S (2017) Erratum: “Swimming in a two-dimensional brinkman fluid: computational modeling and regularized solutions ” [Phys Fluids 28, 021902 (2016)]. Phys Fluids 29:029901

    Google Scholar 

  43. Leshansky A (2009) Enhanced low-Reynolds-number propulsion in heterogenous viscous environments. Phys Rev E 80:051911

    Google Scholar 

  44. Lindemann C, Macauley L, Lesich K (2005) The counterbend phenomenon in dynein-disabled rat sperm flagella and what it reveals about the interdoublet elasticity. Biophys J 89:1165–1174

    Google Scholar 

  45. Mettot C, Lauga E (2011) Energetics of synchronized states in three-dimensional beating flagella. Phys Rev E 84:061905-1–14

    Google Scholar 

  46. Miradbagheri S, Fu H (2016) Helicobacter pylori couples motility and diffusion to actively create a heterogeneous complex medium in gastric diseases. Phys Rev Lett 116:198101

    Google Scholar 

  47. Moore H, Dvorakova K, Jenkins N, Breed W (2002) Exceptional sperm cooperation in the wood mouse. Nature 418:174–177

    Google Scholar 

  48. Morandotti M (2012) Self-propelled micro-swimmers in a Brinkman fluid. J Biol Dyn 6:88–103

    MathSciNet  MATH  Google Scholar 

  49. Mortimer S (2000) CASA-practical aspects. J Androl 21:515–524

    Google Scholar 

  50. Neal C, Hall-McNair A, Kirkman-Brown J, Smith D, Gallagher M (2020) Doing more with less: the flagellar end piece enhances the propulsive effectiveness of human spermatozoa. Phys Rev Fluids 5:073101

    Google Scholar 

  51. Nelson B, Kaliakatsos I, Abbott J (2010) Microrobots for minimally invasive medicine. Annu Rev Biomed Eng 12:55–85

    Google Scholar 

  52. Nganguia H, Pak O (2018) Squirming motion in a Brinkman medium. J Fluid Mech 855:554–573

    MathSciNet  MATH  Google Scholar 

  53. Novati G, Mahadevan L, Koumoutsakos P (2019) Controlled gliding and perching through deep-reinforcement-learning. Phys Rev Fluids 4:093902

    Google Scholar 

  54. Olson S, Fauci L (2015) Hydrodynamic interactions of sheets vs. filaments: attraction, synchronization, and alignment. Phys Fluids 27:121901

    MATH  Google Scholar 

  55. Olson S, Leiderman K (2015) Effect of fluid resistance on symmetric and asymmetric flagellar waveforms. J Aero Aqua Bio-mech 4(1):12–17

    Google Scholar 

  56. Olson S, Suarez S, Fauci L (2011) Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model. J Theor Biol 283:203–216

    MATH  Google Scholar 

  57. Omori T, Ishikawa T (2019) Swimming of spermatozoa in a maxwell fluid. Micromachines 10:78

    Google Scholar 

  58. Pelle D, Brokaw C, Lindemann C (2009) Mechanical properties of the passive sea urchin sperm flagellum. Cell Motil Cytoskeleton 66:721–735

    Google Scholar 

  59. Peskin C (2002) The immersed boundary method. Acta Numer 11:459–517

    MathSciNet  MATH  Google Scholar 

  60. Plouraboue F, Thiam EI, Delmotte B, Climent E (2017) Identification of internal properties of fibres and micro-swimmers. Proc R Soc A 473:20160517

    MathSciNet  MATH  Google Scholar 

  61. Raissi M, Yazdani A, Karniadakis GE (2020) Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367(6481):1026–1030. https://doi.org/10.1126/science.aaw4741

    Article  Google Scholar 

  62. Riedel I, Kruse K, Howard J (2005) A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309:300–303

    Google Scholar 

  63. Riedel-Kruse I, Hilfinger A, Howard J, Julicher F (2007) How molecular motors shape the flagellar beat. HFSP J 1:192–208

    Google Scholar 

  64. Rutllant J, Lopez-Bejar M, Lopez-Gatius F (2005) Ultrastructural and rheological properties of bovine vaginal fluid and its relation to sperm motility and fertilization: a review. Reprod Dom Anim 40:79–86

    Google Scholar 

  65. Saltzman WM, Radomsky ML, Whaley KJ, Cone RA (1994) Antibody diffusion in human cervical mucus. Biophys J 66:508

    Google Scholar 

  66. Sanders L (2009) Microswimmers make a splash: tiny travelers take on a viscous world. Sci News 176:22–25

    Google Scholar 

  67. Sauzade M, Elfring G, Lauga E (2012) Taylor’s swimming sheet: analysis and improvement of the perturbation series. Physica D 240:1567–1573

    MATH  Google Scholar 

  68. Schoeller S, Keaveny E (2018) Flagellar undulations to collective motion: predicting the dynamics of sperm suspensions. J R Soc Interface 15:20170834

    Google Scholar 

  69. Simons J, Fauci L, Cortez R (2015) A fully three-dimensional model of the interaction of driven elastic filaments in a Stokes flow with applications to sperm motility. J Biomech 48:1639–1651

    Google Scholar 

  70. Smith D, Gaffney E, Blake J, Kirkman-Brown J (2009a) Human sperm accumulation near surfaces: a simulation study. J Fluid Mech 621:289–320

    MATH  Google Scholar 

  71. Smith D, Gaffney E, Gadêlha H, Kapur N, Kirkman-Brown J (2009b) Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity. Cell Motil Cytoskel 66(4):220–236

    Google Scholar 

  72. Spielman L, Goren SL (1968) Model for predicting pressure drop and filtration efficiency in fibrous media. Environ Sci Technol 1(4):279–287

    Google Scholar 

  73. Stuart A (2010) Inverse problems: a Bayesian perspective. Acta Numer 19:451–559

    MathSciNet  MATH  Google Scholar 

  74. Suarez S (2010) How do sperm get to the egg? Bioengineering expertise needed!. Exp Mech 50:1267–1274

    Google Scholar 

  75. Suarez S, Pacey A (2006) Sperm transport in the female reproductive tract. Hum Reprod Update 12:23–37

    Google Scholar 

  76. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Philadelphia

    Google Scholar 

  77. Taylor G (1951) Analysis of the swimming of microscopic organisms. Proc R Soc Lond Ser A 209:447–461

    MathSciNet  MATH  Google Scholar 

  78. Taylor G (1952) The action of waving cylindrical tails in propelling microscopic organisms. Proc R Soc Lond Ser A 211:225–239

    MathSciNet  MATH  Google Scholar 

  79. Teran J, Fauci L, Shelley M (2010) Viscoelastic fluid response can increase the speed of a free swimmer. Phys Rev Lett 104:038101–4

    Google Scholar 

  80. Thomases B, Guy R (2014) Mechanisms of elastic enhancement and hindrance for finite-length undulatory swimmers in viscoelastic fluids. Phys Rev Lett 113:098102. https://doi.org/10.1103/PhysRevLett.113.098102

    Article  Google Scholar 

  81. Tierno P, Golestanian R, Pagonabarraga I, Sagues F (2008) Magnetically actuated colloidal micro swimmers. J Phys Chem B 112:16525–16528

    Google Scholar 

  82. Tokic G, Yue D (2012) Optimal shape and motion of undulatory swimming organisms. Proc Biol Sci 279:3065–3074

    Google Scholar 

  83. Tsang A, Tong P, S N, Pak O (2019) Self-learning how to swim at low Reynolds number. arXiv:1808.07639

  84. Vanik MW, Beck JL, Au SK (2000) Bayesian probabilistic approach to structural health monitoring. J Eng Mech 126(7):738–745

    Google Scholar 

  85. Woolley D, Crockett R, Groom W, Revell S (2009) A study of synchronisation between the flagella of bull spermatozoa, with related observations. J Exp Biol 212:2215–2223

    Google Scholar 

  86. Xu G, Wilson K, Okamoto R, Shao J, Dutcher S, Bayly P (2016) Flexural rigidity and shear stiffness of flagella estimated from induced bends and counterbends. Biophys J 110:2750–2768

    Google Scholar 

  87. Yang Y, Elgeti J, Gompper G (2008) Cooperation of sperm in two dimensions: synchronization, attraction, and aggregation through hydrodynamic interactions. Phys Rev E 78:061903-1–9

    Google Scholar 

Download references

Acknowledgements

Simulations were run at the Center for Computation and Visualization at Brown University. KL and AM were partially supported by the NSF through grants DMS-1521266 and DMS-1552903. SDO was supported, in part, by NSF grant DMS-1455270.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sarah D. Olson.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix

Details on Micro-swimmer Simulations

As shown in (4) and (5), the forces that the swimmer exerts on the surrounding fluid are based on a variational derivative of the energy, which has several components. The bending energy of the flagellum is

$$\begin{aligned} E^{j}_{F,\mathrm{bend}}=K^j_{C}\int _{\Gamma _{F}^{j}}\left( \zeta ^j(s,t)-{\hat{\zeta }}^j(s,t)\right) ^2\mathrm{d}s, \end{aligned}$$
(13)

where \(\Gamma _F^j\) is the centerline curve corresponding to the jth flagellum and \(K_{C}\) is a stiffness coefficient enforcing the curvature or bending constraint. The preferred curvature \({\hat{\zeta }}^j\), based on (6), and actual curvature \(\zeta ^j(s,t)\) of the flagellum are given as

$$\begin{aligned} {\hat{\zeta }}^j=\frac{\partial ^2 {\hat{y}}^j}{\partial s^2},\zeta ^j(s,t)= \frac{\frac{\partial ^2 y^j}{\partial s^2}\frac{\partial x^j}{\partial s} - \frac{\partial ^2 x^j}{\partial s^2}\frac{\partial y^j}{\partial s}}{\left( \left( \frac{\partial x^j}{\partial s}\right) ^2 + \left( \frac{\partial y^j}{\partial s}\right) ^2\right) ^{3/2}}, \end{aligned}$$
(14)

where \(\varvec{X}_F^j=[x^j,y^j]\), the portion of \(\varvec{X}\) corresponding to the tail.

In addition to the bending component, we will account for an additional energy component that will tend to maintain the inextensibility of the flagellum. This results in

$$\begin{aligned} E_{F,\mathrm{tens}}^j=\int _{\Gamma _F^j}K_{T}^j\left( \left| \left| \frac{\partial ^2\varvec{X}_F^j}{\partial s^2}\right| \right| -1\right) ^2\mathrm{d}s, \end{aligned}$$
(15)

which, in a discretized form, corresponds to Hookean springs between points on the flagellum with stiffness coefficient \(K_{T}\).

Similar to the flagellum, we assume a preferred shape or curvature of the head. In this simple model, we will assume a head shape with radius \(H_r\) and preferred curvature \({\hat{\kappa }}=1/H_r\). The corresponding energy is

$$\begin{aligned} E_{H,\mathrm{bend}}^j=\int _{\Gamma _H^j}K_{H,C}^j\left( \kappa ^j(s,t)-{\hat{\kappa }}(s,t)\right) ^2\mathrm{d}s, \end{aligned}$$

where \(\Gamma _H^j\) corresponds to the circular head. Here, the actual curvature \(\kappa ^j(s,t)\) is calculated using the same equation as \(\zeta \) in (14), but now \(\varvec{X}_H^j=[x^j,y^j]\), the portion of \(\varvec{X}\) corresponding to the head. In addition, we also have an energy to maintain inextensibility in the head, the same as (15) using \(\Gamma _H^j\), \(\varvec{X}_H^j\), and \(H_{C,\mathrm{tens}}^j\), where we envision Hookean springs between points on the membrane of the head as well as springs connecting points on the circular head that are \(\pi \) apart (we choose \({\mathcal {N}}_H\), the number of points on the head, to be even to ensure points and springs exactly \(\pi \) apart).

The swimmer is initialized (left to right) to have the center of the circular head be placed with a y-coordinate the same as the rightmost point on the flagellum and an x-coordinate that is shifted to the right of the rightmost point by \(H_r\) and an additional small distance apart, dN. To ensure that the passive head remains attached to the actively bending flagellum, and to represent the stiff neck region of a sperm, we connect the head and flagellum with five springs. These springs connect the rightmost point (the \({\mathcal {N}}_F\)th point) of the flagellum to the points on the circle with \(\theta =(\pi -\mathrm{d}\theta ),\pi ,(\pi +\mathrm{d}\theta )\) where \(\mathrm{d}\theta \) is the angular spacing between the \({\mathcal {N}}_H\) points on the head. Additionally, there are two springs connecting the second rightmost point on the flagellum (\({\mathcal {N}}_F-1\)) to the points on the circle with \(\theta =\pi \pm \mathrm{d}\theta \). These springs will have a stiffness coefficient \(K_{N,\mathrm{tens}}\). There is also an energy based on the desired angle between the flagellum and the head. Let \(\mathbf {z}_1\) be the vector connecting the \({\mathcal {N}}_F\)th point on the flagellum and the point on the head with \(\theta =\pi \) and let \(\mathbf {z}_2\) be the vector connecting the points on the head with \(\theta =\pi \pm \mathrm{d}\theta \). In general, we wish for these vectors to be approximately orthogonal, and we can derive an energy and hence forces that penalize this deviation, tending to maintain \(\varvec{z}_1\cdot \varvec{z}_2=0\) with stiffness coefficient \(K_{N,\mathrm{ang}}\) (Fauci and McDonald 1995).

Table 6 Parameters for swimmer model

Given a configuration for each of the \({\mathcal {M}}_S\) swimmers at the initial time point, we determine the forces on the \({\mathcal {M}}_S{\mathcal {N}}_T\) discretized points using (4), where each of the components are calculated using (13)–(15). Second order finite difference approximations are utilized in the calculation of all derivatives in the energy components and a trapezoidal rule is used to calculate the integrals. The forces are then used to calculate the resulting velocity at points along the discretized swimmer, (3b). The location of the swimmer is updated using the no-slip condition, numerically implemented with a forward Euler method. The next time step is reached, where this calculation is repeated.

In these simulations, when there is more than one swimmer, we assume that the beat form parameters such as the amplitude and beat frequency are the same for each swimmer. In addition, we assume that all stiffness parameters are the same. All parameters are given in Table 6, previously benchmarked on experimental data and asymptotic swimming speeds (Ho et al. 2016, 2019; Leiderman and Olson 2016; Olson et al. 2011).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Larson, K., Olson, S.D. & Matzavinos, A. A Bayesian Framework to Estimate Fluid and Material Parameters in Micro-swimmer Models. Bull Math Biol 83, 23 (2021). https://doi.org/10.1007/s11538-020-00852-6

Download citation

Keywords

  • Micro-swimmer
  • Brinkman
  • Bayesian
  • Parameter estimation
  • Stokes
  • Fluid–structure interaction

Mathematics Subject Classification

  • 74F10
  • 92C19
  • 62F15
  • 65C20