This document is a technical report intended to support understanding, use, and development of the codebase fro the Conformation & Concentration Dependent Drag (C2D2) model for viscoelastic stresses in unentangled polymer solutions. It is a working draft: notation, derivations, algorithmic details, and default numerical settings may evolve as the software matures. Portions of the model description and primary equations may later be adapted into one or more journal manuscripts; those manuscripts will cite a versioned, archived copy of this report and the corresponding code release.

Hydrodynamic interactions (HI) between segments of a polymer have long been known to strongly affect polymer stretching in laminar viscometric flows. Yet the role of HI in fluctuating turbulent flows remains unclear. Using Brownian dynamics simulations, we examine the stretching dynamics of bead-spring chains with interbead HI, as they are transported in a homogeneous isotropic turbulent flow (within the ultradilute, one-way coupling regime). We find that the effects of HI are negligibly small for dumbbells but become increasingly prominent as the polymer model is refined, i.e., as the number of beads are increased. HI-endowed chains exhibit a steeper coil-stretch transition as the elastic relaxation time is increased, i.e., HI cause less stretching of stiff polymers and more stretching of moderately elastic polymers. The probability distribution function of the end-to-end extension is also modified, with HI significantly limiting the range of extensions over which a power-law range appears. On quantifying the repeated stretching and recoiling of chains by computing persistence time distributions, we find that HI delays migration between stretched and coiled states. These effects of HI, which are consistent with chains experiencing an effective conformation-dependent drag, are sensitive to the level of coarse graining in the bead-spring model. Specifically, an HI-endowed dumbbell, which cannot form a physical coil, is unable to experience the hydrodynamic shielding effect of HI. Our results highlight the importance of incorporating an extension-dependent drag force in dumbbell-based simulations of turbulent polymer solutions. To develop and test such an augmented dumbbell model, we propose the use of a time-correlated Gaussian random flow, in which the turbulent stretching statistics are shown to be well approximated.

Eukaryotic cilia and flagella exhibit complex beating patterns that change depending on environmental conditions such as fluid viscosity. The mechanism behind these beating pattern transitions remains unclear. We show here that such transitions may arise via a universal elastohydrodynamic mechanism, without the need for changes in dynein forcing or regulation within individual cells. We simulate Kirchhoff rods driven internally by a bending-moment wave that is unidirectional in the material frame, which results in both planar and non-planar beating depending on the growth of twist perturbations at different physical parameters. Changing the viscosity parameter while keeping the internal driving parameters fixed results in beat transitions comparable to experimentally observed transitions in bull and sea urchin sperm when the medium viscosity is varied. Mapping simulation results to transitional viscosities observed in our experiments suggests an assay to estimate the average dynein force in individual cells, with potential diagnostic applications for sperm health.

Capillary thinning of a liquid bridge is an exemplar of complex flow, where the macroscopic geometry couples tightly to the microscopic evolution of polymer conformations. Since its inception, capillary-breakup rheometry (CBR) has been viewed as a tool for measuring a single relaxation time. Yet experiments show that the apparent relaxation time depends systematically on polymer concentration, device geometry, and the preparation protocol. We argue that this variability is not a flaw, but evidence that thinning should be treated as a benchmark complex flow for testing constitutive models. We recast the output of a CBR experiment as the self-selected elastic strain rate, expressed through the elastic Weissenberg number Wi_e, rather than an apparent relaxation time, and organize it in an elastocapillary Pipkin diagram – Wi_e against a geometry-controlled Deborah number. A single-mode, mid-filament stress balance yields a family of Pipkin curves with universal features – a low-De_0 plateau and a finite-extensibility-constrained rise – that a scaling analysis collapses onto a master curve, with an elastic-onset-referenced Deborah number absorbing the unmeasured initial prestretch. The Conformation- and Concentration-Dependent Drag (C2D2) model, acting through coil-stretch hysteresis, lowers the plateau below the Entov-Hinch value and organizes data spanning decades in molecular weight and concentration, across a range of devices, where the classical FENE-P model cannot. The Pipkin diagram framework offers a path toward master plots for classes of polymer solutions, clarifying what is universal in extension-dominated flows.

Difficulties in extracting rheological properties from capillary breakup extensional rheometry (CaBER) are addressed by using Brownian dynamics (BD) simulations of bead-spring polymer chains with finite extensibility (FE), excluded volume (EV), and hydrodynamic interactions (HIs), coupled with a simplified multistage description of flow kinematics. The deformation dependence of HIs is modeled by the approximate “C2D2” model [Prabhakar et al., Phys. Rev. Fluids 2(1), 011301 (2017)], which uses a stretch- and concentration-dependent bead-drag coefficient, thereby avoiding the unmanageably large expense of solving the BD equations with the Rotne–Prager tensor for long chains that influence each other hydrodynamically. We validate our simulations with literature data for polyethylene oxide solutions in both step strain and slow retraction CaBER. We obtained the kinematics of CaBER by combining an analytical early time viscous solution for the filament radius versus time for Newtonian filaments with a later-time balance of capillary stress, viscous stress, and polymer stress. The polymer stress is derived from BD simulations, updating the extension rate as a function of time using the stress balance. The success of this method allows us to clarify the complex relationship between the longest equilibrium polymer relaxation time τZ and the apparent relaxation time τa inferred from the CaBER fiber diameter using D(t)∝exp(−t/3τa) in the regime of exponential decay of diameter with time. We find computationally that τa can be either greater than or less than τZ since FE reduces τa and HI increases it, to extents that depend on molecular weight, polymer concentration, endplate radii, and endplate separation protocol. Maps of EV, FE, and HI dependencies, molecular weight, and the size of endplate diameter relationship are presented, which helps clarify the rheology measured by CaBER and provides rules of thumb to interpret the data extracted from CaBER for experimentalists.

We investigate the assembly of monodisperse polymer-coated polystyrene colloids during the evaporation of sessile drops on oil-coated substrates. In our system, the particles rapidly adsorb at the air-water interface, and as evaporation proceeds, those left behind in the shrinking droplet, are immediately captured by the moving interface. In sufficiently dilute dispersions, nearly all particles adsorb well before the interface becomes saturated, leading to the formation of monolayer sheets. In contrast, at higher particle concentrations, the interface saturates while a significant fraction of particles remain in the bulk, preventing monolayer formation. At any particle concentration, the thickness of the oil layer relative to the initial drop size plays a key role in determining whether monolayers form. The thinnest oil layers provide the broadest range of particle concentrations for successful monolayer assembly. A minimal theoretical model, based on rapid adsorption and negligible influence of internal flows, predicts the boundary between monolayer and non-monolayer regimes with good agreement to experiment. This suggests that interface saturation-not capillary flow-is the controlling mechanism in these drop-on-oil systems. Finally, analysis of the hexatic order parameter reveals that crystalline domain size increases with particle number. These findings offer a pathway to tune monolayer structure and order through controlled drying protocols, with potential applications in 2D materials and functional coatings.

The viscoelastic properties of the female reproductive tract influence sperm swimming behavior, but the exact role of these rheological changes in regulating sperm energetics remains unknown. Using high-speed dark-field microscopy, the flagellar dynamics of free-swimming sperm across a physiologically relevant range of viscosities is resolved. A transition from 3D to 2D slither swimming under an increased viscous loading is revealed, in the absence of any geometrical or chemical stimuli. This transition is species-specific, aligning with viscosity variations within each species' reproductive tract. Despite substantial drag increase, 2D slithering sperm maintain a steady swimming speed across a wide viscosity range (20-250 and 75-1000 mPa s for bull and human sperm) by dissipating over sixfold more energy into the fluid without elevating metabolic activity, potentially by altering the mechanisms of dynein motor activity. This energy-efficient motility mode is ideally suited for the viscous environment of the female reproductive tract. The viscoelastic properties of the female reproductive tract influence sperm migration, yet the understanding of how this complex rheological environment impacts sperm flagellar dynamics and energetics is limited. Here, a transition from 3D to 2D slither swimming under increased viscous loading is revealed, in the absence of geometrical or chemical stimuli, revealing species-specific transition ranges and improved energy efficiency.image

Sheet-like colloidal assemblies represent model systems to investigate the structure and properties of two-dimensional materials. Here, we report a simple yet versatile method for the preparation of colloidal monolayer sheet-like assemblies that affords control over the size, crystalline order, flexibility, and defect density. The protocol that we report relies on self-assembly of colloids as a sessile drop of dispersion is evaporated on an oil-covered substrate. In this case, the contact line continually moves as the drop shrinks. Polyethyleneimine polymer-covered micrometer-sized colloidal particles are transported to the air-water interface and assemble to form a monolayer sheet as the drop dries. Cross-linking the polymer renders the colloidal assembly permanent. Interestingly, monodisperse colloidal particles form disordered assemblies when dried from low concentration dispersions, while polycrystalline ordered assemblies form at higher concentrations. We demonstrate that increasing the cross-linker to polymer ratio decreases the flexibility of the assembly. Introduction of different-sized colloidal particles in a sheet leads to increased disorder. Removal of sacrificial particles from the sheet allowed the introduction of "holes" in the sheets. Thus, these colloidal sheets are models for probing the effects of disorder, doping, and vacancies in two-dimensional systems.

We demonstrate through the use of a unique acoustically driven microfluidic extensional rheometry platform (ADMiER) that a single measurement – i.e. the time required for a liquid bridge filament comprising a microlitre semen sample to thin and break up under elastocapillary stresses – constitutes an appropriate proxy for quantifying the motile sperm concentration of the sample in place of computer-assisted sperm analysis (CASA) and haemocytometer measurements used in conventional semen assessment – without the need to separately resolve for individual dependencies on each sperm parameter. By benchmarking diagnostic test accuracy results of blind random bull semen samples ( $n=35$ ) against OpenCASA measurements of these parameters, ADMiER is capable of predicting sperm quality to 93.7 % accuracy, 91.4 % sensitivity and 97.5 % specificity, with respect to commonly adopted veterinary industry minimum values for fertility. These results therefore highlight the potential diagnostic capability of the platform as a conceptual first step towards the development of a rapid, low-cost and portable alternative for veterinary male bovine fertility assessment.

Complex rheological properties of the female reproductive tract serve as a long-range guidance mechanism for sperm. However, conventional microscopy methods and population-level studies have restricted our understanding of the combined effects of flow and viscosity on sperm flagellar beating behavior at the single-cell level. Here, using microfluidics, we study individual sperm flagellar dynamics within a physiologically relevant range of viscosity and shear rate. Our results indicate that an increase in shear rate from 0 to 6 s-1 at 75 mPa s reduces the flagellar curvature by 20%, while the peak energy occurs at a shear rate of 3 s-1-a favorable shear rate for sperm rheotaxis. Viscosity has a more dominant influence than shear rate on the flagellar waveform, and the synergistic effect of flow and viscosity promotes energy-efficient beating behavior. Our findings provide new insights into the complex interplay between the environment of the female reproductive tract and sperm function in facilitating fertilization.

Cell mechanics and motility are responsible for collective motion of cells that result in overall deformation of epithelial tissues. On the other hand, contact-dependent cell-cell signalling is responsible for generating a large variety of intricate, self-organized, spatial patterns of the signalling molecules. Moreover, it is becoming increasingly clear that the combined mechanochemical patterns of cell shape/size and signalling molecules in the tissues, for example, in cancerous and sensory epithelium, are governed by mechanochemical coupling between chemical signalling and cell mechanics. However, a clear quantitative picture of how these two aspects of tissue dynamics, i.e., signalling and mechanics, lead to pattern and form is still emerging. Although, a number of recent experiments demonstrate that cell mechanics, cell motility, and cell-cell signalling are tightly coupled in many morphogenetic processes, relatively few modeling efforts have focused on an integrated approach. We extend the vertex model of an epithelial monolayer to account for contact-dependent signalling between adjacent cells and between non-adjacent neighbors through long protrusional contacts with a feedback mechanism wherein the adhesive strength between adjacent cells is controlled by the expression of the signalling molecules in those cells. Local changes in cell-cell adhesion lead to changes in cell shape and size, which in turn drives changes in the levels of signalling molecules. Our simulations show that even this elementary two-way coupling of chemical signalling and cell mechanics is capable of giving rise to a rich variety of mechanochemical patterns in epithelial tissues. In particular, under certain parametric conditions, bimodal distributions in cell size and shape are obtained, which resemble experimental observations in cancerous and sensory tissues.

Capillary thinning of liquid bridges is routinely used for extensional rheology of Newtonian and complex fluids. Although it is expected that the volume and aspect ratio of a liquid bridge significantly influence its dynamics, the role played by these parameters in rheological characterization has not been previously studied. We perform numerical simulations of Newtonian as well as viscoelastic liquid bridges with the one-dimensional slender-filament approximation of Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and Ardekani et al. [“Dynamics of bead formation, filament thinning and breakup in weakly viscoelastic jets,” J. Fluid Mech. 665, 46–56 (2010)]. Sample volume and bridge aspect ratio control two phenomena that can adversely impact rheological characterization: the tendency to form satellite drops at the necking plane and the slowing down of capillary thinning due to the proximity (in parameter space) of the liquid-bridge stability boundary. The optimal range of these parameter values to avoid drop formation and slowdown is discussed.

Sperm swim through the female reproductive tract by propagating a 3D flagellar wave that is self‐regulatory in nature and driven by dynein motors. Traditional microscopy methods fail to capture the full dynamics of sperm flagellar activity as they only image and analyze sperm motility in 2D. Here, an automated platform to analyze sperm swimming behavior in 3D by using thin‐lens approximation and high‐speed dark field microscopy to reconstruct the flagellar waveform in 3D is presented. It is found that head‐tethered mouse sperm exhibit a rolling beating behavior in 3D with the beating frequency of 6.2 Hz using spectral analysis. The flagellar waveform bends in 3D, particularly in the distal regions, but is only weakly nonplanar and ambidextrous in nature, with the local helicity along the flagellum fluctuating between clockwise and counterclockwise handedness. These findings suggest a nonpersistent flagellar helicity. This method provides new opportunities for the accurate measurement of the full motion of eukaryotic flagella and cilia which is essential for a biophysical understanding of their activation by dynein motors.

Abstract Study question How does viscosity influence the flagellar beating behaviour of free-swimming bull, mouse, and human sperm? Summary answer Sperm flagellar beating behaviour exhibits a transition from an irregular three-dimensional (3D) beating at 5 mPa·s to an organized two-dimensional (2D) waveform at 20 mPa·s. What is known already Sperm migrate in a complex viscoelastic environment through the female reproductive tract. The viscoelastic properties of the oviductal fluid significantly influence the progressive motility of sperm, acting as one of the key guidance mechanisms in vivo. However, the biomechanics of sperm flagellar activity in response to varying viscosity of the oviductal fluid is poorly understood. Understanding sperm flagellar behaviour in physiologically relevant environments is crucial to understanding reproduction and may help to describe unknown causes of infertility. Lack of high-speed high-resolution imaging techniques and automated image-analysis capabilities have been the main barriers to fully describe the flagellar beating behaviour. Study design, size, duration We used a custom-built high-speed high-resolution dark-field microscopy platform to resolve the flagellar dynamics of human, bull, and mouse sperm near surfaces in viscoelastic media ranging in viscosity from 1 to 250 mPa·s. The imaging system includes an automated image analysis algorithm to quantify sperm flagellar waveform and motility characteristics by extracting the flagellar centreline, reconstructing the waveform and calculating tangent-angle profiles. 20 sperm from 3 different bull, mice and humans were analyzed. Participants/materials, setting, methods Bull, mouse, and human sperm were used in this study. In each experiment, a diluted sperm sample in a buffer supplemented with methylcellulose was used and free-swimming sperm were imaged using dark-field microscopy at 200 frames per second. An automated image analysis algorithm was used to extract sperm flagellar centreline and Proper Orthogonal Decomposition (POD) was then used to study sperm flagellar waveform. Statistical analysis was performed using one-way ANOVA. Main results and the role of chance The reconstructed flagellar beating pattern was different for sperm swimming in low and high-viscosity media. Bull sperm exhibited a lower flagellar beating amplitude along the end piece when swimming in a high-viscosity media, a potential energy-efficient strategy to navigate a high viscosity fluid. The first two dominant POD modes (shape modes) describe more than 90% of the beating pattern for all species. Bull sperm exhibited a transition mode with irregular loops in 5 mPa·s buffer, but the flagellar shape cycle created an organised repetitive circular cycle in 1 mPa.s (3D beating) buffer and at viscosities higher than 5 mPa·s (2D beating). Human sperm also indicated a similar behaviour but with the transition happening at higher viscosities. Mouse sperm in high-viscosity media had a lower flagellar beating amplitude across the principal piece and higher beating amplitude across the end piece. The flagellar shape cycle in mouse sperm showed a periodic flagellar beating behaviour at high-viscosities (>20 mPa·s), but a shape cycle with distorted loops at lower viscosities. Our results showed in quantitative detail that increasing viscosity alters sperm flagellar beating pattern, and how sperm migration behaviour in low viscosity media can be distinct from their swimming behaviour in vivo. Limitations, reasons for caution A more comprehensive study of sperm motility parameters such as curvilinear velocity, average path velocity and straight line velocity with a larger sample size is required to fully characterise sperm swimming behaviour as a function of viscosity. Wider implications of the findings The increasing viscosity of the oviductal fluid regulates the sperm flagellar beating behaviour to switch from a 3D swimming behaviour with irregular shape cycles at lower viscosities to a 2D slithering mode with repetitive circular shape cycles at higher viscosities to achieve a more energy-efficient beating pattern for navigation. Trial registration number Not applicable

Fertilization requires sperm to travel long distances through the complex environment of the female reproductive tract. Despite the strong association between poor motility and infertility, the kinetics of sperm tail movement and the role individual proteins play in this process is poorly understood. Here, we use a high spatiotemporal sperm imaging system and an analysis protocol to define the role of CRISPs in the mechanobiology of sperm function. Each of CRISP1, CRISP2, and CRISP4 is required to optimize sperm flagellum waveform. Each plays an autonomous role in defining beat frequency, flexibility, and power dissipation. We thus posit that the expansion of the CRISP family from one member in basal vertebrates, to three in most mammals, and four in numerous rodents, represents an example of neofunctionalization wherein proteins with a common core function, boosting power output, have evolved to optimize different aspects of sperm tail performance.

The colonisation of a soft passive material by motile cells such as bacteria is common in biology. The resulting colonies of the invading cells are often observed to exhibit intricate patterns whose morphology and dynamics can depend on a number of factors, particularly the mechanical properties of the substrate and the motility of the individual cells. We use simulations of a minimal 2D model of self-propelled rods moving through a passive compliant medium consisting of particles that offer elastic resistance before being plastically displaced from their equilibrium positions. It is observed that the clustering of active (self-propelled) particles is crucial for understanding the morphodynamics of colonisation. Clustering enables motile colonies to spread faster than they would have as isolated particles. The colonisation rate depends non-monotonically on substrate stiffness with a distinct maximum at a non-zero value of substrate stiffness. This is observed to be due to a change in the morphology of clusters. Furrow networks created by the active particles have a fractal-like structure whose dimension varies systematically with substrate stiffness but is less sensitive to particle activity. The power-law growth exponent of the furrowed area is smaller than unity, suggesting that, to sustain such extensive furrow networks, colonies must regulate their overall growth rate.

We demonstrate a technique for investigating the energetics of flagella or cilia. We record the planar beating of tethered mouse sperm at high resolution. Beating waveforms are reconstructed using proper orthogonal decomposition of the centerline tangent-angle profiles. Energy conservation is employed to obtain the mechanical power exerted by the dynein motors from the observed kinematics. A large proportion of the mechanical power exerted by the dynein motors is dissipated internally by the motors themselves. There could also be significant dissipation within the passive structures of the flagellum. The total internal dissipation is considerably greater than the hydrodynamic dissipation in the aqueous medium outside. The net power input from the dynein motors in sperm from Crisp2-knockout mice is significantly smaller than in wildtype samples, indicating that ion-channel regulation by cysteine-rich secretory proteins controls energy flows powering the axoneme.

A key challenge in biology is to understand how spatiotemporal patterns and structures arise during the development of an organism. An initial aggregate of spatially uniform cells develops and forms the differentiated structures of a fully developed organism. On the one hand, contact-dependent cell-cell signalling is responsible for generating a large number of complex, self-organized, spatial patterns in the distribution of the signalling molecules. On the other hand, the motility of cells coupled with their polarity can independently lead to collective motion patterns that depend on mechanical parameters influencing tissue deformation, such as cellular elasticity, cell-cell adhesion and active forces generated by actin and myosin dynamics. Although modelling efforts have, thus far, treated cell motility and cell-cell signalling separately, experiments in recent years suggest that these processes could be tightly coupled. Hence, in this paper, we study how the dynamics of cell polarity and migration influence the spatiotemporal patterning of signalling molecules. Such signalling interactions can occur only between cells that are in physical contact, either directly at the junctions of adjacent cells or through cellular protrusional contacts. We present a vertex model which accounts for contact-dependent signalling between adjacent cells and between non-adjacent neighbours through long protrusional contacts that occur along the orientation of cell polarization. We observe a rich variety of spatiotemporal patterns of signalling molecules that is influenced by polarity dynamics of the cells, relative strengths of adjacent and non-adjacent signalling interactions, range of polarized interaction, signalling activation threshold, relative time scales of signalling and polarity orientation, and cell motility. Though our results are developed in the context of Delta-Notch signalling, they are sufficiently general and can be extended to other contact dependent morpho-mechanical dynamics.