Movement is an essential feature of life. It allows organisms to move towards a more favorable environment and to search for food. There are many biological systems that fall under the category active matter, from molecular motors walking on microtubules inside cells to flocks of birds. What these systems have in common is that each of its constituents converts energy into directed motion, that is, they propel themselves forward. Besides the many biological examples, there is also synthetic active matter, these are self-propelled particles made in a laboratory. These are typically colloidal sized particles that can propel themselves forward by self-phoresis. In this work the focus is on the low Reynolds number regime, meaning that the typical size of the constituents is less than a few micrometers. The models that are used to describe such active matter are can be viewed as nonequilibrium extensions to Brownian motion (the thermal motion of small particles dissolved in a fluid). In many systems the self-propulsion speed (activity) is not homogeneous in space: the particles swim faster in some areas than in others. The main topic of this dissertation is how a single active particle, or a few active particles tied together by a potential, behave in such systems. It is known that a single active particle without any steering mechanism spends most time in the regions where it moves slowly, or in other words, they spend most time in regions where they are less active. However, here it is shown that, even though they spend most time in the less active regions, dynamical properties, such as the probability to move towards the more active regions is higher than moving towards the less active regions. Furthermore, when the active particles are connected to a passive Brownian 'cargo' particle, chained together to form a colloidal sized polymer, or fixed to another active particle, the resulting active dimers or polymers either accumulate in the high activity regions or the low activity regions, depending on the friction of the cargo particle, the number of monomers in the polymer, or the relative orientation of active particles. Lastly, when the activity is both time- and space-dependent, a steady drift of active particles can be induced, without any coupling between the self-propulsion direction and the gradient in the activity. This phenomenon can be used to position the particles depending on their size.:1. Brownian Motion 2. Active Matter 3. Modeling Active Matter 4. Introduction: Inhomogeneous activity 5. Pseudochemotaxis 6. Cargo-Carrying Particles 7. Active Colloidal Molecules 8. Time-Varying Activity Fields Appendix: Hydrodynamics

The development of an organism starting from a fertilized egg involves the self-organized formation of patterns and the generation of shape. Patterns and shapes are characterized by their geometry, i.e. angles and distances between features. In this thesis, we set out to understand how the given geometry of pattern and shape of a living system feeds back into the evolution of this geometry. We focus on two fundamental developmental processes: axis specification and gastrulation. Both processes rely on the directed movements of cells and molecules driven by molecular force generation. Here, we ask how the geometry of an embryo guides such active flows. Active flows are often confined to the surface of a cell or embryo which is usually curved. We use the hydrodynamic theory of active surfaces to investigate how this curvature impacts on flows that are driven by patterns of mechanical activity. Using a minimal model of the cell cortex, we find that active cortical stresses can drive a rotation of the cell that aligns the chemical pattern of the stress regulator with the geometry of the cell surface. In particular, we find that active tension in the cytokinetic ring ensures that a cell divides along its longest axes, a common phenomenon known as Hertwig’s rule. As a consequence, the body axes of the C. elegans embryo are aligned with the geometry of the egg shell. We next set out to understand the impact of surface geometry on flows and patterns in more complex geometries. We focus in particular on localized sources of mechanical activity in curved fluid films. Such active particles act as sensors of the surface geometry, as the viscosity relates the local flow field to the large-scale geometry of the fluid film. We find that the impact of an anisotropic surface geometry on the flow field can generally be understood in terms of effective gradients of friction and viscosity. With this, we show that contractile points in a fluid film are attracted by protrusions and saddle geometries where the contractile point is surrounded by a maximal amount of surface area within the hydrodynamic length. Furthermore, we find that anisotropic active particles move towards or away from a saddle of the surface depending on whether they are extensile or contractile. To understand the process of gastrulation and left-right symmetry breaking in the avian embryo, we develop a hydrodynamic theory of the primitive streak, a line of mechanically active material. With this theory of an active viscous crack, we analyze experimental data from quail embryos. We find that the embryo-scale cell movements during gastrulation are driven by mechanical activity at the streak, while the surrounding epithelium behaves like a homogeneous fluid film. With this mechanical model, we find that streak elongation does not require extensile forces along the streak. Instead, streak elongation results from the flux of tissue into the streak, the viscosity of the surrounding tissue and the polar geometry of the streak. During avian left-right symmetry breaking, a chiral flow of tissue emerges at the tip of the streak, the so called Hensen’s node. We find that this flow results from an active torque that drives a counter-rotation of tissue layers. Thus, avian left-right symmetry breaking is facilitated by the mechanical coupling of tissue layers that the structure of node and streak provides. Finally, we study how the geometry of a surface impacts on such chiral flows. We find that chiral flows at the avian node as well as in the cell cortex can be recapitulated as the result of molecular torque dipoles that are aligned with the tangential plane of the cell or tissue surface. Only when the surface is curved, such in-plane torques drive in-plane flows. Thus, the geometry of the avian node and the cytokinetic furrow may facilitate the chiral flows that are driven by these structures. Taken together, we find that the geometry of an embryo is crucial to the flows and patterns that emerge in such a mechanically active system, because the geometry defines how forces and torques are transmitted.:1 Introduction 1.1 Embryogenesis from a geometric viewpoint 1.2 Hydrodynamic theory of active fluid films 1.3 Understanding active surfaces with complex numbers 1.4 Overview of this thesis 2 Crack mechanics of avian gastrulation 2.1 The primitive streak as a crack in a fluid film 2.2 Hydrodynamic theory of active viscous cracks 2.3 The primitive streak as a branch cut 2.4 Advective crack propagation 2.5 Discussion 3 Pattern formation guided by surface geometry 3.1 Minimal model of guided symmetry breaking 3.2 Diffusion on a curved surface 3.3 Pattern formation in an active fluid model of the cell cortex 3.4 Discussion 4 Geometry sensing by active flows 4.1 Geometry sensing by an active isotropic fluid 4.2 Deformation response of active flow in general surface geometries 4.3 Geometry sensing by a contractile point 4.4 Pattern formation guided by the geometric potential 4.5 Geometry sensing by active p-atic particles 4.6 Discussion 5 Chiral flows controlled by embryo geometry 5.1 Mechanical model of avian left-right symmetry breaking 5.2 Chiral flows facilitated by curvature gradients 5.3 Discussion 6 Conclusion and Outlook

Biological matter is driven far from thermodynamic equilibrium by active processes on the molecular scale. These processes are usually driven by the chemical reaction of a fuel and generate spontaneous movements and mechanical stresses in the system, even in the absence of external forces or torques. Moreover these active stresses effectively fluidify the material. The cell cytoskeleton, suspensions of swimming microorganisms or tissues are prominent examples of active fluids. Active processes in biological systems often exhibit chiral asymmetries. Examples are the chirality of cytoskeletal filaments which interact with motor proteins, the chirality of the beat of cilia and flagella as well as the helical trajectories of many biological micro-swimmers. Moreover, large scale chiral flows have been observed in the cell cortex of C. elegans and Xenopus embryos. Active force generation induces force and torque dipoles in the material. If all forces are internal the total force and torque vanish as required by the conservation of momentum and angular momentum. The density of force dipoles is an active stress in the material. In addition, active chiral processes allow for the existence of active torque dipoles which enter the conservation of angular momentum and generate an active antisymmetric stress and active angular momentum fluxes. We developed a generic description of active fluids that takes into account active chiral processes and explicitly keeps track of spin and orbital angular momentum densities. We derived constitutive equations for an active chiral fluid based on identifying the entropy production rate from the rate of change of the free energy and linearly expanding thermodynamic fluxes in terms of thermodynamic forces. We identified four elementary chiral motors that correspond to localized distributions of chiral force and torque dipoles that differ by their symmetry and produce different chiral fluid flows and intrinsic rotation fields. We employ our theory to analyze different active chiral processes. We first show that chiral flows can occur spontaneously in an active fluid even in the absence of chiral processes. For this we investigate the Taylor-Couette motor, that is an active fluid confined between two concentric cylinders. For sufficiently high active stresses the fluid generates spontaneous rotations of the two cylinders with respect to each other thus breaking the chiral symmetry of the system spontaneously. We then investigate cases where active chiral processes on the molecular scale break the chiral symmetry of the whole system. We show that chiral flows occur in films of chiral motors and derive a generic theory for thin films of active fluids. We discuss our results in the context of carpets of beating cilia or E. coli swimming close to a surface. Finally, we discuss chiral flows that are observed in the cellular cortex of the nematode C. elegans at the one cell stage. Two distinct chiral flow events are observed. The first chiral flow event (i) is a screw like chiral rotation of the two cell halves with respect to each other and occurs around 15min after fertilization. This event coincides with the establishment of cortical cell polarity. The second chiral flow event (ii) is a chiral rotation of the entire cell cortex around the anterior posterior axis of the whole cell and occurs around 30min after fertilization. Measuring densities of molecular motors during episode (i) we fit the flow patterns observed using only two fit parameters: the hydrodynamic length and cortical chirality. The flows during (ii) can be understood assuming an increase of the hydrodynamic length. We hypothesize that the cell actively regulates the cortical viscosity and the friction of the cortex with the eggshell and cytosol. We show that active chiral processes in soft biological matter give rise to interesting new physics and are essential to understand the material properties of many biological systems, such as the cell cortex.

Fire ants link their legs and jaws together to form functional structures called self- assemblages. Examples include floating rafts, towers, bridges, and bivouacs. We investigate these self-assemblages of fire ants. Our studies are motivated in part by the vision of providing guidance for programmable robot swarms. The goal for such systems is to develop a simple programmable element from which complex patterns or behaviors emerge on the collective level. Intelligence is decentralized, as is the case with social insects such as fire ants. In this combined experimental and theoretical study, we investigate the construction of two fire ant self-assemblages that are critical to the colony’s survival: the raft and the tower. Using time-lapse photography, we record the construction processes of rafts and towers in the laboratory. We identify and characterize individual ant behaviors that we consistently observe during assembly, and incorporate these behaviors into mathematical models of the assembly process. Our models accurately predict both the assemblages’ shapes and growth patterns, thus providing evidence that we have identified and analyzed the key mechanisms for these fire ant self-assemblages. We also develop novel techniques using scanning electron microscopy and micro-computed tomography scans to visualize and quantify the internal structure and packing properties of live linked fire ants. We compare our findings to packings of dead ants and similarly shaped granular material packings to understand how active arranging affects ant spacing and orientation. We find that ants use their legs to increase neighbor spacing and hence reduce their packing density by one-third compared to packings of dead ants. Also, we find that live ants do not align themselves in parallel with nearest neighbors as much as dead ants passively do. Our main contribution is the development of parsimonious mathematical models of how the behaviors of individuals result in the collective construction of fire ant assemblages. The models posit only simple observed behaviors based on local information, yet their mathe- matical analysis yields accurate predictions of assemblage shapes and construction rates for a wide range of ant colony sizes.

[pt] Organismos vivos têm seus próprios meios de locomoção e são capazes de se reproduzir. Além disto, o habitat no qual os organismos estão inseridos é tipicamente heterogêneo, de modo que as condições ambientais variam no tempo e no espaço. Nesta tese, são propostos e investigados modelos teóricos para compreender o comportamento coletivo de organismos vivos, visando responder questões relevantes sobre a organização e preservação da população utilizando técnicas analíticas e numéricas. Inicialmente, considerando um habitat homogêneo, em que as propriedades estatísticas das condições ambientais são independentes do tempo e do espaço, estudamos como padrões espaço-temporais podem emergir na distribuição da população devido a interações não-locais e investigamos o papel das flutuações ambientais neste processo. Em seguida, assumindo um meio ambiente heterogêneo, analisamos o caso de um único domínio de habitat. Considerando uma classe de equações não lineares, introduzindo flutuações temporais e interações entre os organismos, fornecemos uma perspectiva geral da estabilidade de populações neste caso, desafiando os conceitos ecológicos anteriores. Em um segundo passo, assumindo uma paisagem complexa fragmentada, consideramos que os indivíduos têm acesso a informações sobre a estrutura espacial do meio. Mostramos que os indivíduos sobrevivem quando as regiões espaciais viáveis estão suficientemente aglomeradas e observamos que o tamanho da população é maximizado quando os indivíduos utilizam parcialmente a informação do meio ambiente. Finalmente, como resultados exatos analíticos não são factíveis em muitas situações importantes, propomos uma abordagem efetiva para interpretar os dados experimentais. Assim, somos capazes de conectar a heterogeneidade do ambiente e a persistência da população, caracterizada pela distribuição de probabilidade para os tempos de vida. / [en] Living entities have their own means of locomotion and are capable of reproduction. Furthermore, the habitat in which organisms are embedded is typically heterogeneous, such that environment conditions vary in time and space. In this thesis, theoretical models to understand the collective dynamics of living beings have been proposed and investigated aiming to address relevant questions such as population organization and persistence in the environment, using analytical and numerical techniques. Initially, considering an homogeneous habitat, in which the statistical properties of the environmental conditions are time and space independent, we study how spatiotemporal order can emerge in the population distribution due to nonlocal interactions and investigate the role of environment fluctuations in the self-organization process. Further, we continue our investigation assuming an heterogeneous environment, starting with the simplest case of a single habitat domain, and we obtain the critical conditions for population survival for different population dynamics. Considering a class of nonlinear equations, introducing temporal oscillations and interactions among the organisms, we are able to provide a general picture of population stability in a single habitat domain, challenging previous ecological concepts. At last, assuming a fragmented complex landscape, resembling realistic properties observed in nature, we additionally assume that individuals have access to information about the spatial structure. We show that individuals survive when patches of viable regions are clustered enough and, counter-intuitively, observe that population size is maximized when individuals have partial information about the habitat. Finally, since, analytical exact results are not feasible in many important situations, we propose an effective approach to interpret experimental data. This way we are able to connect environment heterogeneity and population persistence.

Originalmente introduzidas por T. Vicksek et al. [Phys. Rev. Lett. 75, 1226 (1995)], partÂıculas auto-propelidas (PAP) possuem uma velocidade intrÂınseca constante que sofre variaÂcËoes em sua direÂcËao como resultados de perturbaÂcËoes externas (outras partÂıculas ou meio) e sËao usadas para modelar sistemas que apresentam efeitos de aglomeraÂcËao. O conceito de PAP Âe aplicado para descrever e entender efeitos dinËamicos de aglomeraÂcËao em sistemas naturais, tais como microorganismos vivos (bactÂerias, vÂırus, etc) e colËonias de indivÂıduos de que se movem em bandos (peixes, ovelhas, abelhas, etc) ou, produzidos artificialmente, como sistemas coloidais especialmente preparadas em laboratÂorio. O estudo de PAP tem sua relevËancia em diversas Âareas do conhecimento, tais como engenharia de materiais, medicina e ciËencias da natureza (fÂısica, quÂımica e biologia). Na maioria dos casos, o movimento coletivo tem um comportamento bastante diferenciado dos movimentos individuais dos componentes de um dado sistema. Assim, o movimento de um certo indivÂıduo Âe influenciado pela presenÂca dos outros constituintes do sistema, alterando o seu comportamento geral, como consequËencia da interaÂcËao direta entre eles. Desta forma, vemos a importËancia da investigaÂcËao e entendimento do comportamento coletivo das PAP. Nesta dissertaÂcËao, estudamos um sistema bidimensional binÂario de PAP na presenÂca de obstÂaculos rÂıgidos com geometria anisotrÂopica (semi-cÂırculos) distribuÂıdos na forma de uma rede quadrada. AlÂem das interaÂcËoes partÂıcula-partÂıcula e partÂıcula-obstÂaculo, o movimento individual de cada PAP sofre influËencia de um ruÂıdo branco. O objetivo Âe caracterizar o transporte de PAP atravÂes do substrato bidimensional na ausËencia de uma forÂca externa propulsora. Apresentamos um estudo sistemÂatico do movimento coletivo das PAP em funÂcËao das velocidades das partÂıculas, da intensidade do ruÂıdo que define o movimento estocÂastico das PAP, do tamanho dos obstÂaculos, da densidade de PAP e da separaÂcËao entre os obstÂaculos. Devido a anisotropia dos obstÂaculos, surge um movimento coletivo espontËaneo e ordenado na direÂcËao normal `a superfÂıcie plana dos obstÂaculos, caracterizado por uma velocidade mÂedia nËao-nula para cada tipo de PAP na ausËencia de forÂca externa e que Âe influenciado pelos parËametros do sistema / Originally introduced by T. Vicksek et al. [Phys. Rev. Lett. 75, 1226 (1995)], Self Propelled Particles (SPP) have an intrinsic constant speed which suffer variations ins its direction as results of external perturbations (another particles or system) and are used to model systems that shows agglomeration effects. The concept of SPP is applied to describe and understand dynamic effects of agglomeration in natural systems, such as living micro-organisms (bacteria, virus, etc.) and colonies of individuals which move in flocks (fishes, sheep, bees) or, artificially produced, as colloidal systems especially prepared in laboratory. The study of SPP has its relevance in several areas of knowledge, such as material engineering, medicine and sciences of nature (physics, chemistry and biology). In most of cases, the collective motion has an well-differentiated behaviour of the individual motion of the components of a given system. So, the movement of a certain individual is affected by the presence of the other elements of the system, changing its general behaviour, as direct consequences of the direct interaction between them. In this way, we see the importance of investigation and understanding of collective motion of the SPP. Especially in this dissertation, we study a binary two-dimensional system of SPP subject to the presence of rigid obstacles with anisotropic geometry (semi-circles) distributed neatly in form of a square web. Beyond the particle-particle and particleobstacle interaction, the individual movement of each SPP suffers influence of an white noise. The objective is characterize the transport of SPP trough the two-dimensional substratum in absence of an propeller external force. We present and systematic study of collective motion of SPP in function of the speed of the particles, of the noise intensity which defines the stochastic movement of SPP, of the size of the obstacles, of the SPP density e the separation between the obstacles. Due the anisotropy of the obstacles, arise an spontaneous and ordered collective motion in normal direction of the plane surface of the obstacles, characterized by an non-null mean speed for each type of SPP in absence of an external force which in affected by the system parameters.

L'émergence de mouvements cohérents à grande échelle a été abondamment observée dans les populations animales (nuées d'oiseaux, bancs de poissons, essaims de bactéries...) et plus récemment au sein de systèmes artificiels. De tels ensembles d'individus auto-propulsés, susceptibles d'aligner leurs vitesses, présentent des propriétés physiques singulières. Cette thèse théorique étudie divers aspects de ces systèmes actifs polaires.Dans un premier temps, nous avons modélisé une population de colloïdes auto-propulsés. En étroite association avec les travaux expérimentaux, nous avons décrit la dynamique du niveau individuel à l'échelle macroscopique. Les résultats théoriques expliquent l'émergence et la structure de motifs cohérents : (i) transition vers le mouvement collectif, (ii) propagation de structures spatiales polarisées, (iii) amortissement des fluctuations de densité dans un liquide polaire, (iv) vortex hétérogène dans des géométries confinées.D'un point de vue plus fondamental, nous avons ensuite étudié les excitations non linéaires qui se propagent dans les systèmes actifs polaires. L'analyse des théories hydrodynamiques de la matière active, à l'aide d'outils issus des systèmes dynamiques, a permis de rationaliser les observations expérimentales et numériques reportées jusqu'ici.Enfin, nous avons proposé une approche complémentaire pour caractériser les populations actives. Associant étude numérique et résultats analytiques, nous avons étudié les propriétés géométriques des trajectoires individuelles, ainsi que leur enchevêtrement au sein de groupes tridimensionnels. Ces observables pourraient permettre de sonder efficacement la dynamique de populations animales. / The emergence of coherent motion at large scale has been widely observed in animal populations (bird flocks, fish schools, bacterial swarms...) and more recently in artificial systems. Such ensembles of self-propelled individuals, capable of aligning their velocities, are commonly referred to as polar active materials. They display unique physical properties, which we investigate in this theoretical thesis.We first describe a population of self-propelled colloids. In strong connection with the experiments, we model the dynamics from the individual level to the macroscopic scale. The theoretical results account for the emergence and the structure of coherent patterns: (i)~transition to collective motion, (ii)~propagation of polar spatial structures, (iii)~damping of density fluctuations in a polar liquid, (iv)~heterogeneous vortex in confined geometries.We then follow a more formal perspective, and study the non-linear excitations which propagate in polar active systems. We analyze the hydrodynamic theories of active matter using a dynamical-system framework. This approach makes it possible to rationalize the experimental and numerical observations reported so far.Finally, we propose a complementary approach to characterize active populations. Combining numerical and analytical results, we study the geometric properties of the individual trajectories and their entanglement within three-dimensional flocks. We suggest that these observables should provide powerful tools to describe animal flocks in the wild.

No description available.

In tissue development, wound healing and aberrant cancer progression cell–cell interactions drive mixing and segregation of cellular composites. However, the exact nature of these interactions is unsettled. Here we study the dynamics of packed, heterogeneous cellular systems using wound closure experiments. In contrast to previous cell sorting experiments, we find non-universal sorting behavior. For example, monolayer tissue composites with two distinct cell types that show low and high neighbor exchange rates (i.e., MCF-10A & MDA-MB-231) produce segregated domains of each cell type, contrary to conventional expectation that the construct should stay jammed in its initial configuration. On the other hand, tissue compounds where both cell types exhibit high neighbor exchange rates (i.e., MDA-MB-231 & MDA-MB-436) produce highly mixed arrangements despite their differences in intercellular adhesion strength. The anomalies allude to a complex multi-parameter space underlying these sorting dynamics, which remains elusive in simpler systems and theories merely focusing on bulk properties. Using cell tracking data, velocity profiles, neighborhood volatility, and computational modeling, we classify asymmetric interfacial dynamics. We indicate certain understudied facets, such as the effects of cell death & division, mechanical hindrance, active nematic behavior, and laminar & turbulent flow as their potential drivers. Our findings suggest that further analysis and an update of theoretical models, to capture the diverse range of active boundary dynamics which potentially influence self-organization, is warranted.

Active matter represents a unique class of non-equilibrium systems, including examples ranging from cellular structures to large-scale biological tissues. These systems exhibit intriguing spatiotemporal dynamics, driven by the constituent particles’ continuous energy expenditure. Such active-matter systems, featuring complex hydrodynamics, are described by sophisticated mathematical models, typically using partial differential equations (PDEs). PDEs modeling hydrodynamics, such as the Navier-Stokes equations, are analytically intractable, and notoriously challenging to study computationally. The challenges include the need for consistent numerical methods along with their efficient and scalable high-performance computer implementation to solve the PDEs numerically. However, when considering new theoretical PDE models, such as active hydrodynamics, conventional approaches often fall short due to the specialization made in the numerical methods to study certain specific models. The inherent complexity and nonlinearity of active-matter PDEs add to the challenge. Hence, the computational study of such active-matter PDE models requires rapidly evolving high-performance computer software that can easily implement new numerical methods to solve these equations in biologically realistic three-dimensional domains. This presents a rich, yet underexplored territory demanding scalable computational frameworks that apply to a large class of PDEs. In this thesis, we introduce a computational framework that effectively allows for using multiple numerical methods through a context-aware template expression system akin to an embedded domain-specific language. This framework primarily aims at solving lengthy PDEs associated with active hydrodynamics in complex domains, while experimenting with new numerical methods. Existing PDE-solving codes often lack this flexibility, as they are closely tied to a PDE and domain geometry that rely on a specific numerical method. We overcome these limitations by using an object-oriented implementation design, and show experiments with adaptive and numerically consistent particle-based approach called Discretization-Corrected Particle Strength Exchange (DC-PSE). DC-PSE allows for the higher-order discretization of differential operators on arbitrary particle distributions leading to the possibility of solving active hydrodynamic PDEs in complex domains. However, the curse of dimensionality makes it difficult to numerically solve three-dimensional equations on single-core architectures and warrants the use of parallel and distributed computers. We design a novel template-expression system and implement it in the scalable scientific computing library OpenFPM. Our methodology offers an expression-based embedded language, enabling PDE codes to be written in a form that closely mirrors mathematical notation. Leveraging OpenFPM, this approach also ensures parallel scalability. To further enhance our framework's versatility, we employ a \textit{separation-of-concerns} abstraction, segregating the model equations from numerics, and domain geometry. This allows for the rapid rewriting of codes for agile numerical experiments across different model equations in various geometries. Supplementing this framework, we develop a distributed algebra system compatible with OpenFPM and Boost Odeint. This algebra system opens avenues for a multitude of explicit adaptive time-integration schemes, which can be selected by modifying a single line of code while maintaining parallel scalability. Motivated by symmetry-preserving theories of active hydrodynamics, and as a first benchmark of our template-expression system, we present a high-order numerically convergent scheme to study active polar fluids in arbitrary three-dimensional domains. We derive analytical solutions in simple Cartesian geometries and use them to show the numerical convergence of our algorithm. Further, we showcase the scalability of the computer code written using our expression system on distributed computing systems. To cater to the need for solving PDEs on curved surfaces, we present a novel meshfree numerical scheme, the Surface DC-PSE method. Upon implementation in our scalable framework, we benchmark Surface DC-PSE for both explicit and implicit Laplace-Beltrami operators and show applications to computing mean and Gauss curvature. Finally, we apply our computational framework to exploring the three-dimensional active hydrodynamics of biological flowing matter, a prominent model system to study the active dynamics of cytoskeletal networks, celluar migration, and tissue mechanics. Our software framework effectively tackles the challenges associated to numerically solving such non-equilibrium spatiotemporal PDEs. We perform linear perturbation analysis of the three-dimensional Ericksen-Leslie model and find an analytical expression for the critical active potential or, equivalently, a critical length of the system above which a spontaneous flow transition occurs. This spontaneous flow transition is a first realization of a three-dimensional active Fr\'eedericksz transition. With our expression system, we successfully simulate 3D active fluids, finding phases of spontaneous flow transitions, traveling waves, and spatiotemporal chaos with increasing active stress. We numerically find a topological phase transition similar to the Berezinskii–Kosterlitz–Thouless transition (BKT transition) of the two-dimensional XY model that occurs in active polar fluids after the spontaneous flow transition. We then proceed to non-Cartesian geometries and show the application of our software framework to solve the active polar fluid equations in spherical domains. We find spontaneous flows in agreement with recent experimental observations. We further showcase the framework to solve the equations in 3D annular domains and a `peanut' geometry that resembles a dividing cell. Our simulations further recapitulate the actin flows observed in \textit egg extracts within spherical shell geometries, showcasing our framework's versatility in handling complex geometrical modifications of model equations. Looking ahead, we hope our framework will serve as a foundation for further advancements in computational morphogenesis, fostering collaboration and using the present techniques in biophysical modeling.