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.
Similar content being viewed by others
Change history
10 February 2021
Funding information was corrected
References
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
Auriault JL (2009) On the domain of validity of Brinkman’s equation. Trans Porous Media 79:215–223
Beck JL, Yuen KV (2004) Model selection using response measurements: Bayesian probabilistic approach. J Eng Mech 130(2):192–203
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
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
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
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
Cortez R (2001) The method of regularized Stokeslets. SIAM J Sci Comput 23:1204–1225
Cortez R, Cummins B, Leiderman K, Varela D (2010) Computation of three-dimensional Brinkman flows using regularized methods. J Comput Phys 229:7609–7624
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
Dillon R, Fauci L, Yang X (2006) Sperm motility and multiciliary beating: an integrative mechanical model. Comput Math Appl 52:749–758
Durlofsky L, Brady JF (1987) Analysis of the Brinkman equation as a model for flow in porous media. Phys Fluids 30(11):3329–3341
Elfring G, Lauga E (2009) Hydrodynamic phase locking of swimming microorganisms. Phys Rev Lett 103:088101
Elfring G, Lauga E (2011a) Passive hydrodynamic synchronization of two-dimensional swimming cells. Phys Fluids 23:011902
Elfring G, Lauga E (2011b) Synchronization of flexible sheets. J Fluid Mech 674:163–173
Elfring G, Pak O, Lauga E (2010) Two-dimensional flagellar synchronization in viscoelastic fluids. J Fluid Mech 646:505–515
Elgeti J, Kaupp U, Gompper G (2010) Hydrodynamics of sperm cells near surfaces. Biophys J 99(4):1018–1026
Elgeti J, Winkler R, Gompper G (2015) Physics of microswimmers—single particle motion and collective behavior: a review. Rep Prog Phys 78:056601
Fauci L, McDonald A (1995) Sperm motility in the presence of boundaries. Bull Math Biol 57:679–699
Flemming H, Wingender J (2010) The biofilm matrix. Nat Rev Microbiol 8:623–633
Fu H, Powers T, Wolgemuth C (2007) Theory of swimming filaments in viscoelastic media. Phys Rev Lett 99:258101–05
Fu H, Wolgemuth C, Powers T (2009) Swimming speeds of filaments in nonlinearly viscoelastic fluids. Phys Fluids 21:033102
Fu H, Shenoy V, Powers T (2010) Low Reynolds number swimming in gels. Europhys Lett 91:24002
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
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
Gallagher M, Smith D, Kirkman-Brown J, Cupples G (2020) Fast. https://www.flagellarcapture.com. Accessed 29 Nov 2020
Gao W, Wang J (2014) Synthetic micro/nanomotors in drug delivery. Nanoscale 6:10486–10494
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
Ho N, Leiderman K, Olson S (2016) Swimming speeds of filaments in viscous fluids with resistance. Phys Rev E 93(4):043108
Ho N, Leiderman K, Olson S (2019) A 3-dimensional model of flagellar swimming in a Brinkman fluid. J Fluid Mech 864:1088–1124
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
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
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
Ishimoto K, Gaffney E (2018b) Hydrodynamic clustering of human sperm in viscoelastic fluids. Sci Rep 8:15600
Jeznach C, Olson S (2020) Dynamics of swimmers in fluids with resistance. Fluids 5(14):1–20
Kaipio J, Somersalo E (2005) Statistical and computational inverse problems, vol 160. Springer, Berlin
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
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
Lauga E (2007) Propulsion in a viscoelastic fluid. Phys Fluids 19:083104
Lauga E, Powers T (2009) The hydrodynamics of swimming microorganisms. Rep Prog Phys 72:096601
Leiderman K, Olson S (2016) Swimming in a two-dimensional Brinkman fluid: computational modeling and regularized solutions. Phys Fluids 28(2):021902
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
Leshansky A (2009) Enhanced low-Reynolds-number propulsion in heterogenous viscous environments. Phys Rev E 80:051911
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
Mettot C, Lauga E (2011) Energetics of synchronized states in three-dimensional beating flagella. Phys Rev E 84:061905-1–14
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
Moore H, Dvorakova K, Jenkins N, Breed W (2002) Exceptional sperm cooperation in the wood mouse. Nature 418:174–177
Morandotti M (2012) Self-propelled micro-swimmers in a Brinkman fluid. J Biol Dyn 6:88–103
Mortimer S (2000) CASA-practical aspects. J Androl 21:515–524
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
Nelson B, Kaliakatsos I, Abbott J (2010) Microrobots for minimally invasive medicine. Annu Rev Biomed Eng 12:55–85
Nganguia H, Pak O (2018) Squirming motion in a Brinkman medium. J Fluid Mech 855:554–573
Novati G, Mahadevan L, Koumoutsakos P (2019) Controlled gliding and perching through deep-reinforcement-learning. Phys Rev Fluids 4:093902
Olson S, Fauci L (2015) Hydrodynamic interactions of sheets vs. filaments: attraction, synchronization, and alignment. Phys Fluids 27:121901
Olson S, Leiderman K (2015) Effect of fluid resistance on symmetric and asymmetric flagellar waveforms. J Aero Aqua Bio-mech 4(1):12–17
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
Omori T, Ishikawa T (2019) Swimming of spermatozoa in a maxwell fluid. Micromachines 10:78
Pelle D, Brokaw C, Lindemann C (2009) Mechanical properties of the passive sea urchin sperm flagellum. Cell Motil Cytoskeleton 66:721–735
Peskin C (2002) The immersed boundary method. Acta Numer 11:459–517
Plouraboue F, Thiam EI, Delmotte B, Climent E (2017) Identification of internal properties of fibres and micro-swimmers. Proc R Soc A 473:20160517
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
Riedel I, Kruse K, Howard J (2005) A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309:300–303
Riedel-Kruse I, Hilfinger A, Howard J, Julicher F (2007) How molecular motors shape the flagellar beat. HFSP J 1:192–208
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
Saltzman WM, Radomsky ML, Whaley KJ, Cone RA (1994) Antibody diffusion in human cervical mucus. Biophys J 66:508
Sanders L (2009) Microswimmers make a splash: tiny travelers take on a viscous world. Sci News 176:22–25
Sauzade M, Elfring G, Lauga E (2012) Taylor’s swimming sheet: analysis and improvement of the perturbation series. Physica D 240:1567–1573
Schoeller S, Keaveny E (2018) Flagellar undulations to collective motion: predicting the dynamics of sperm suspensions. J R Soc Interface 15:20170834
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
Smith D, Gaffney E, Blake J, Kirkman-Brown J (2009a) Human sperm accumulation near surfaces: a simulation study. J Fluid Mech 621:289–320
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
Spielman L, Goren SL (1968) Model for predicting pressure drop and filtration efficiency in fibrous media. Environ Sci Technol 1(4):279–287
Stuart A (2010) Inverse problems: a Bayesian perspective. Acta Numer 19:451–559
Suarez S (2010) How do sperm get to the egg? Bioengineering expertise needed!. Exp Mech 50:1267–1274
Suarez S, Pacey A (2006) Sperm transport in the female reproductive tract. Hum Reprod Update 12:23–37
Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Philadelphia
Taylor G (1951) Analysis of the swimming of microscopic organisms. Proc R Soc Lond Ser A 209:447–461
Taylor G (1952) The action of waving cylindrical tails in propelling microscopic organisms. Proc R Soc Lond Ser A 211:225–239
Teran J, Fauci L, Shelley M (2010) Viscoelastic fluid response can increase the speed of a free swimmer. Phys Rev Lett 104:038101–4
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
Tierno P, Golestanian R, Pagonabarraga I, Sagues F (2008) Magnetically actuated colloidal micro swimmers. J Phys Chem B 112:16525–16528
Tokic G, Yue D (2012) Optimal shape and motion of undulatory swimming organisms. Proc Biol Sci 279:3065–3074
Tsang A, Tong P, S N, Pak O (2019) Self-learning how to swim at low Reynolds number. arXiv:1808.07639
Vanik MW, Beck JL, Au SK (2000) Bayesian probabilistic approach to structural health monitoring. J Eng Mech 126(7):738–745
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
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
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
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
Authors and Affiliations
Corresponding author
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
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
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
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
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).
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
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-020-00852-6