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[EQUATION] where the second equality holds by Corollary 10.4 13. Reasonably closed Wadge classes In this section we will define reasonably closed Wadge classes and prove that every good Wadge class is reasonably closed. This notion is an ad hoc definition, and it is the key idea of an ingenious lemma due to Harrington ... |
Given [MATH] , set [EQUATION] Notice that every element of [MATH] is obtained by alternating finite blocks of zeros and finite blocks of ones. Define the function [MATH] by setting |
[EQUATION] where we start counting with the [MATH] block of zeros. It is easy to check that [MATH] is continuous. Definition 13.1 |
Let [MATH] be a Wadge class in [MATH] . We will say that [MATH] is reasonably closed if [MATH] for every [MATH] The following result is essentially the same as vE3 , Lemma 4.2.17] , except that it is not limited to the Borel context. |
Lemma 13.2 Assume [MATH] . Let [MATH] be a good Wadge class in [MATH] . Then [MATH] is reasonably closed. Proof. By Corollary 8.2 , we can fix [MATH] such that [MATH] . Set [MATH] . Pick [MATH] . Notice that [MATH] by Lemma 6.4 . Therefore, by Lemma 6.4 , there exists [MATH] such that [MATH] . Since [MATH] is a good Wa... |
14. Wadge classes of homogeneous spaces are good The main result of this section is that [MATH] is a good Wadge class whenever [MATH] is a homogeneous space of sufficiently high complexity (see Theorem 14.4 for the precise statement). Together with Lemma 13.2 , this will allow us to apply Theorem 15.1 in the next secti... |
We will need three preliminary results. Lemmas 14.1 14.2 , and 14.3 correspond to vE3 , Lemma 4.2.16] vE3 , Lemma 4.4.2] , and vE3 , Lemma 4.4.1] respectively, while Theorem 14.4 corresponds to vE3 , Lemma 4.4.3] . Once again, the difference is that we work with arbitrary sets instead of just Borel sets. In the case of... |
Lemma 14.1 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] be a good Wadge class in [MATH] . Assume that [MATH] and [MATH] are subspaces of [MATH] such that [MATH] and [MATH] . Then [MATH] |
Proof. Let [MATH] be a homeomorphism. By Ke , Theorem 3.9] , we can fix [MATH] and a homeomorphism [MATH] such that [MATH] . By Corollary 8.2 , we can fix [MATH] such that [MATH] . Notice that [MATH] by Lemma 6.4 . It follows from Lemma 6.4 that [MATH] . Therefore, according to Lemma 6.4 , there exists [MATH] such that... |
Lemma 14.2 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] be a good Wadge class in [MATH] , and let [MATH] be a homogeneous subspace of [MATH] . Assume that [MATH] is non-empty and [MATH] . Then [MATH] |
Proof. Define [MATH] . Notice that [MATH] is a cover of [MATH] because [MATH] is homogeneous and [MATH] is non-empty. Let [MATH] be a countable subcover of [MATH] . Observe that each [MATH] by Lemma 14.1 . Fix [MATH] for [MATH] such that [MATH] for each [MATH] . Set [MATH] for [MATH] , and observe that [MATH] for each ... |
[EQUATION] where [MATH] and [MATH] . In conclusion, we see that [MATH] . Since [MATH] , it follows that [MATH] Lemma 14.3 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] be such that [MATH] , and let [MATH] be codense in [MATH] . Then there exists a non-empty [MATH] and [MATH] suc... |
Proof. Since [MATH] , using Lemma 3.2 and the fact that [MATH] is continuously closed, it is easy to see that [MATH] . Therefore, we can fix [MATH] and pairwise disjoint [MATH] for [MATH] such that [MATH] . Since [MATH] is a Baire space, we can fix [MATH] and a non-empty [MATH] such that [MATH] |
Notice that [MATH] and [MATH] because [MATH] , hence it is possible to apply Proposition 3.5 . In particular, one sees that [MATH] , hence [MATH] . So, we have [MATH] (again by Proposition 3.5 ) and [MATH] . This easily yields the desired result if [MATH] is non-selfdual, so assume that [MATH] is selfdual. By Corollary... |
Theorem 14.4 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] be a homogeneous dense subspace of [MATH] such that [MATH] . Then [MATH] is a good Wadge class in [MATH] |
Proof. Fix [MATH] minimal with respect to the property that [MATH] for some non-empty [MATH] . Fix a non-empty [MATH] such that [MATH] . Assume without loss of generality that [MATH] (the case [MATH] is similar). First we will prove that [MATH] is a good Wadge class, then we will show that [MATH] . Observe that [MATH] ... |
Next, we claim that [MATH] . Assume, in order to get a contradiction, that [MATH] . By Corollary 8.2 , we can fix [MATH] such that [MATH] . Since [MATH] is dense in [MATH] and homogeneous, if [MATH] were countable then [MATH] would be countable, by the same argument as in the proof of Proposition 2.6 . So [MATH] is an ... |
At this point, we know that [MATH] is a good Wadge class, so we can apply Lemma 14.2 , obtaining that [MATH] . To conclude the proof, it will be enough to show that [MATH] is non-selfdual, as it will follow from the minimality of [MATH] and Proposition 3.5 that [MATH] . Assume, in order to get a contradiction, that [MA... |
15. The main results This sections contains our main results. Theorem 15.2 extends (and is inspired by) vE4 , Lemma 2.7] . All the work done so far was aimed at applying the following result, which is a particular case of St2 , Theorem 2] . Given a Wadge class [MATH] in [MATH] and [MATH] , we will say that [MATH] is ev... |
Theorem 15.1 (Steel) Assume [MATH] . Let [MATH] be a reasonably closed Wadge class in [MATH] . Assume that [MATH] and [MATH] are subsets of [MATH] that satisfy the following conditions: |
[MATH] and [MATH] are everywhere properly [MATH] [MATH] and [MATH] are either both meager in [MATH] or both comeager in [MATH] Then there exists a homeomorphism [MATH] such that [MATH] |
Theorem 15.2 Assume [MATH] . Let [MATH] and [MATH] be homogeneous dense subspaces of [MATH] . Assume that [MATH] , and that the following conditions are satisfied: |
[MATH] [MATH] and [MATH] are either both meager spaces or both Baire spaces. Then there exists a homeomorphism [MATH] such that [MATH] |
Proof. Let [MATH] . Notice that [MATH] is a good Wadge class by Theorem 14.4 , hence it is reasonably closed by Lemma 13.2 . It is clear that if [MATH] and [MATH] are both meager spaces, then they are both meager in [MATH] . On the other hand, if [MATH] and [MATH] are both Baire spaces, then they are comeager in [MATH]... |
Corollary 15.3 Assume [MATH] . Let [MATH] be a zero-dimensional homogeneous space that is not locally compact. Then [MATH] is strongly homogeneous. |
Proof. Notice that [MATH] is crowded, otherwise it would be discrete by homogeneity. Therefore, we can assume without loss of generality that [MATH] is a dense subspace of [MATH] . If [MATH] , then the desired result follows from vE3 , Corollary 4.4.6] . So assume that [MATH] |
By Theorem 2.8 , it will be enough to show that [MATH] for every [MATH] . Pick [MATH] . Let [MATH] be a homeomorphism, and let [MATH] . It is easy to check that [MATH] is a homogeneous dense subspace of [MATH] . Furthermore, it is clear that [MATH] and [MATH] are either both meager spaces or both Baire spaces. We claim... |
Set [MATH] , and observe that [MATH] is a good Wadge class by Theorem 14.4 . In particular, [MATH] is non-selfdual. Hence, by Corollary 8.2 , we can fix [MATH] such that [MATH] . Notice that [MATH] by Lemma 6.4 , hence [MATH] by Lemma 6.4 . This shows that [MATH] . In order to prove the other inclusion, by Lemma 3.2 , ... |
# Source: arxiv 1806.00373 # Title: Boundary behaviours of Leishmania mexicana: a hydrodynamic simulation study # Sections: all # Downloaded: 2026-03-02T08:44:14.964784+00:00 |
Boundary behaviours of Leishmania mexicana : a hydrodynamic simulation study (15 November, 2018) Abstract It is well established that the parasites of the genus Leishmania exhibit complex surface interactions with the sandfly vector midgut epithelium, but no prior study has considered the details of their hydrodynamics... |
Leishmania swimming scenarios demonstrate an absence of stable boundary motility for an idealised model promastigote, with behaviours ranging from boundary capture to deflection into the bulk both with and without surface forces between the swimmer and the boundary. Indeed, the inclusion of a short-range repulsive surf... |
keywords: Promastigote motility , Boundary element method , Flagellar beat , Low Reynolds number flow , Leishmania -sandfly gut interaction |
Introduction The unicellular parasitic eukaryotes of the family Trypanosomatidae are the cause of many major human diseases including African trypanosomiasis and New World leishmaniasis |
. Those of the genus Leishmania , transmitted to humans by the bite of a sandfly, affect around 4 million individuals globally . A prominent cause of cutaneous leishmaniasis in the Americas, L. mexicana are a popular focus of recent research owing to their complete development cycle being observable |
in vitro . In the highly motile promastigote stage of their life cycle, a stage defined by morphology and as shown in Fig. , they utilise a single flagellum for locomotion, protruding from their anterior cell body and predominantly beating with a tip-to-base planar wave, the latter being common to all trypanosomatidae |
. Their viability in the sandfly vector midgut is thought to depend upon their ability to navigate effectively , with it being widely accepted that their survival in the low-Reynolds number environment of the sandfly midgut is reliant upon attachment to the nearby epithelium |
. In fact, the precise driving mechanism of the tip-first boundary approach of Leishmania promastigotes remains unknown, and is hypothesised by Bates to simply be a naive consequence of their flagellum-first swimming direction, but the effects of potential hydrodynamic factors remain to be considered in detail. Contras... |
Leishmania -sandfly pairings the mechanism of epithelial binding has been well-explored, evidenced to be dependent upon the major |
Leishmania surface glycoconjugate, lipophosphoglycan (LPG) . Following metacyclogenesis, and an accompanying change in LPG, the epithelial binding is reversed, resulting in the detachment of the promastigote from the midgut surface |
A direct consequence of locomotion via tip-to-base flagellar beating, Leishmania spp. are hydrodynamically classified as pullers achieving propulsion by drawing fluid along the length of the flagellum before then pushing out the fluid at the sides. This is in contrast to |
pushers , such as human spermatozoa and E. coli , which perform the reverse action and are consequently propelled in the opposite direction |
. Differences between the hydrodynamic properties of pushers and pullers have been well documented for the case of squirmers swimmers of nearly constant shape with generated fluid flow at their boundary, a model classically applied to Opalina and other ciliated microorganisms |
. Further, within the classes of pusher and puller fundamentally different behaviours are observed, even for the simplest swimmers, as illustrated by the contrast between a force-dipole puller, which deflects from boundaries |
, and the spherical puller squirmer, which swims stably near boundaries . Hence refined models of cellular swimmers are required to elucidate their boundary dynamics, as illustrated by the rich boundary behaviours observed for flagellate pushers such as E. coli and mammalian spermatozoa, together with the biflagellate ... |
Chlamydomonas , in recent extensive work However, corresponding studies of monoflagellated pullers, either observational or simulation-based, are comparatively lacking and hence there is extensive scope for the investigation of the boundary behaviours of a flagellated puller such as L. mexicana |
In particular, additional to their differing hydrodynamic classification, Leishmania promastigotes are also morphologically distinct from the better-studied pusher monoflagellates. Accumulation behaviours are reported to be sensitive to variations in swimmer morphology |
, even for puller squirmers , while appeal to time reversal symmetry to infer puller behaviour from pusher behaviour requires the same cellular morphology. Hence |
Leishmania swimming behaviour cannot be inferred from previous studies of swimmers, due to its distinct cell morphology, with the lengthscales of the flagellum and cell body approximately equal (at approximately |
[MATH] [MATH] , see Fig. ). In contrast, for a typical human spermatozoon this ratio approaches one-tenth, with the spermatozoon cell body being substantially shorter than the attached flagellum |
The smaller cell body of such spermatozoa also enables the use of approximate analytic techniques such as resistive force theory in studying their motility |
. This has been implemented classically for the spermatozoa of the sea urchin Psammechinus by Gray and Hancock where hydrodynamic interactions between the two cell components are either neglected or treated simplistically. Given the comparable scales of cell body and flagellum in Leishmania , such an approach is inappr... |
, with methods treating the flagellum and cell body comparably being more natural, and indeed, accurate. Thus, a full and high-accuracy numerical study is necessitated to fully capture the hydrodynamics and resulting behaviours of Leishmania |
The functional relevance of a beating flagellum to the promastigote is the control of spatial location. Hence we will examine the mechanics of |
Leishmania upon approach to, and movement away from, a boundary. Thus our aim is to consider how the cell may control its location in the sandfly midgut, in its need to both approach and leave the gut epithelium at different stages of its life cycle. |
Hence, in this paper we will firstly detail digital capture for the flagellum waveform of Leishmania mexicana in a typical growth medium. We then seek a low-dimensional expansion of the observed kinematics via standard Fourier analysis |
, and use a high-accuracy boundary element computational framework to perform a number of in silico experiments , our primary objective being to document the complex long-term behaviour of a virtual promastigote in the presence of a planar boundary. Further, we use beat-averaged phase planes to classify and quantify be... |
Methods 2.1 Videomicroscopy of L. mexicana Leishmania mexicana high framerate videos were generated similarly to previously described in Wheeler . Promastigote L. mexicana (WHO strain MNYC/BZ/62/M379) were grown in M199 supplemented with 10% FCS and 50 [MATH] HEPES [MATH] HCl pH [MATH] , and maintained in exponential g... |
L. mexicana culture in logarithmic growth was added, then a blocked coverslip was added giving a ca. [MATH] [MATH] sample depth. 200 and 400 frame/ [MATH] videos between 4 and 9 [MATH] long were captured with an Andor Neo v5.5 sCMOS camera using phase contrast illumination on a Zeiss Axio Observer inverted microscope w... |
For visualising L. mexicana waveform in the bulk a 250 [MATH] thick adhesive plastic square was applied to a glass slide to make a deep chamber, in which 10 [MATH] |
L. mexicana culture was placed, then a coverslip added. Images were captured at a focal plane mid-way through the sample depth, using dark field illumination and a long working distance [MATH] N.A. 0.45 Plan-Apochromat objective (1063-139). Many cells lay with their cell body and flagellum entirely in the focal plane, ... |
1(c) ). 2.2 Determining flagellar kinematics Flagellar kinematics were extracted via automated analysis in the ImageJ macro language, relative to a cell-fixed reference frame with coordinates |
[MATH] . This frame is defined as having [MATH] directed along the axis joining the body centroid to the visible base of the flagellum, with the base being placed at the origin and having coordinates [MATH] in the inertial laboratory frame (see Fig. |
). In a similar analysis of mammalian spermatozoa , a tangential attachment of the flagellum to the cell body was assumed due to the presence of structural components, such as outer dense fibres, which provided sufficient information to rotate the captured data into the cell-fixed frame. Indeed, this would be appropria... |
. However, our captured data provides an exterior view at resolution such that the perceived flagellar attachment appears free, and at a site that we will refer to as the base, distinct from the true flagellar attachment zone in the flagellar pocket. Therefore, relaxing the constraint of tangential attachment is suitab... |
[MATH] being the centre of rotation here and throughout. Approximate wavelengths and amplitudes were computed similarly to Gadelha et al. where appropriate, along with an approximation to the cell body length. A standard decomposition of the resulting waveform data into Fourier modes was then performed |
, where an expansion of low dimension was sought for use in numerical simulations. 2.3 Governing equations The small scale dynamics of L. mexicana promastigotes in a Newtonian fluid of viscosity [MATH] is governed by the incompressible Stokes equations (see |
Appendix ), with a Reynolds number on the order of [MATH] , using typical length and velocity scales given in Wheeler et al. , Wheeler For a given surface [MATH] , which typically will represent the promastigote surface, we have the following non-dimensional integral representation for the instantaneous flow velocity [... |
[MATH] on the surface, with coordinates given in the body-fixed frame and following Pozrikidis [EQUATION] Here, [MATH] and [MATH] are velocity and stress Green’s functions of 3-dimensional Stokes flow, [MATH] is the surface normal directed into the fluid, [MATH] denotes the surface traction, and [MATH] denotes a princi... |
2.4 The virtual promastigote In order to model the swimming behaviour of Leishmania we introduce a neutrally-buoyant virtual promastigote , which here will have an idealised geometry that is similar to wild-type promastigotes (see |
Figs. and ). In particular, we construct our idealised promastigote using an axisymmetric prolate ellipsoid to represent the cell body, which differs slightly from observed L. mexicana promastigotes, as the latter typically exhibit limited body curvature along their long axis (see |
Fig. for typical examples). With reference to the non-dimensionalisation scales used in Appendix , we prescribe a non-dimensional body length of [MATH] , with circular cross sections of diameter 0.35, consistent with a typical promastigote lengthscale of 10 [MATH] and corresponding to a non-dimensional flagellum length... |
[EQUATION] where the quantity [MATH] is chosen such that the arclength of the flagellum is conserved, and additionally we enforce that |
[MATH] , ensuring flagellar attachment occurs at the same location on the body for all times [MATH] . Where it is appropriate to define a beat plane in the cell-fixed frame, we will assume without loss of generality that such a plane is spanned by unit vectors in the [MATH] -directions, so that there is no beating in t... |
Fig. ). Flagellum material velocities in the inertial frame at a given time [MATH] are approximated from positional information using a central differences scheme optimised for double precision arithmetic |
2.5 Numerical scheme Given the instantaneous velocity of the flagellum in the cell-fixed frame, we proceed to solve the boundary integral equations of 3-dimensional Stokes flow over the discretised virtual promastigote surface, closing the system with the conditions of force and torque–free swimming, which are appropri... |
). We additionally enforce, without loss of generality, that the normal boundary traction has a surface mean of zero, eliminating the pressure non-uniqueness inherent in Stokes flow, and solve the resulting system for the virtual promastigote velocities and surface tractions. Throughout, we use the Blakelet for the int... |
Eq. , along with the accompanying form of [MATH] , which ensures that the solutions satisfy a no-slip condition on a specified planar boundary. Denoting coordinates in the laboratory frame by |
[MATH] , we typically specify this stationary boundary as [MATH] Having computed instantaneous promastigote velocities at a time [MATH] , we use Heun’s method with timestep [MATH] to update the position and orientation of the cell in the inertial frame, as detailed in Smith et al. , with positional evolution of the fla... |
[EQUATION] and cell orientation being dealt with similarly. An adaptive timestepping scheme is employed in order to increase accuracy when approaching the boundary, where velocities are expected to be highly sensitive to boundary separation and body configuration. |
Noting that surface interactions such as steric forces often occur between cells and substrates, but also are highly variable between different solutes and substrates |
, we proceed to additionally consider a surface force. While an attractive surface force will simply tend to induce binding once a cell is sufficiently close to a boundary wall, the impact of a repulsive potential is ambiguous a priori, with the potential to reflect the cell away from the boundary or to induce stable s... |
. The resulting force in non-dimensional form is given by [EQUATION] where [MATH] is the boundary separation, [MATH] is the outward-facing normal of the boundary, and [MATH] are the effective range and strength of the force, chosen such that a strong short-range repulsion is represented, with strength scaling with dime... |
[MATH] and beat period [MATH] Our implementation was verified in free-space against Ishimoto and Gaffney and by reproducing the Jeffery’s orbits of ellipsoidal particles |
, whilst the implementation of the Blakelet was compared with the software library BEMLIB 2.6 Construction of phase planes In an effort to gain a more complete picture of the virtual promastigote dynamics without performing numerous costly individual simulations, we attempt to simplify the dynamics via its restriction ... |
. Specifically, we proceed by equating the third coordinate vectors of the inertial and cell-fixed frames, so that motion and beat plane are confined to the plane [MATH] , without loss of generality. We additionally average over a single beat period, as in |
Ramia et al. , enabling a parameterisation by boundary separation and orientation alone, which we denote by [MATH] and [MATH] respectively (see |
Fig. ). We define the separation to be the distance from the flagellar attachment point to the wall, and the orientation to be the clockwise angle between the body-fixed [MATH] -axis and the boundary normal. We can then form the representation |
[EQUATION] where [MATH] and [MATH] represent the process of boundary element simulation and subsequent phase averaging. Note that we may identify [MATH] and [MATH] , the phase-averaged linear velocity in the [MATH] -direction and the rate of rotation about the [MATH] -axis respectively. |
Results 3.1 L. mexicana exhibit simple flagellar kinematics Analysis of the temporal Fourier spectra of the L. mexicana flagellar beat for a sample of [MATH] cells between two cover slips revealed a single prominent planar beat frequency in the range of 26-34 [MATH] , clearly observable along the entire flagellum lengt... |
4(b) ). The lack of other significant modes suggests a decomposition into a single sinusoid is appropriate for representing the flagellar beat. Thus we opt to define the non-dimensional beat parameters of amplitude, wavelength and frequency, denoted [MATH] [MATH] and [MATH] respectively, and assume the functional form |
[EQUATION] in the cell-fixed frame. Under this assumption, amplitudes and wavelengths were approximated, with the averaged results being shown in |
Fig. 4(a) . Here we set [MATH] and [MATH] for use in simulations, recovering a typical long-wavelength flagellar beat (see Supplementary Movie 1) and noting that the results that follow are not sensitive to variations in these parameter choices. Good agreement between the model flagellar beat and that extracted from da... |
4(c) , demonstrating a remarkably simple flagellar kinematics, not dissimilar to that of L. major and the classically-studied Crithidia oncopelti |
. Additionally, it is noted that whilst this planar beat pattern was seen in confined promastigotes (see Section 2.1 ), such beating is also observed in the bulk and no non-planar beating is exhibited (see Fig. |
1(c) ). Furthermore, from analysing observations of the human spermatozoon, it has also been reported that monoflagellate beating is unchanged near to a boundary to the resolution that can be observed with typical microscopy 23 , Appendix A] Therefore we adopt our model beat pattern both in the far and near-field of bo... |
3.2 Virtual promastigote beat plane aligns towards the perpendicular Long-time simulations of virtual promastigotes revealed a tendency to align their beat plane normal to the boundary when [MATH] , the cell-boundary distance, is sufficiently small. This may be explained by a simple torque balance argument, especially ... |
. In this instance, and with reference to Fig. , the no-slip boundary induces increased drag on the near-side of the cell body in comparison to the far side. This difference results in a torque, which when combined with the constraint of torque-free swimming drives reorientation towards the perpendicular, significantly... |
for various swimmer configurations and body lengthscales. These figures demonstrate the existence of a notable drag difference between near and far sides of the swimmer, and in particular the dependence of this difference on body size, supporting the conclusion that increased cell body size is a factor in the hydrodyna... |
). 3.3 Virtual promastigotes reorient to promote boundary collision via distal flagellar tip Upon a collision-bound approach to a planar boundary, in the absence of additional repulsive surface forces, we observe the remarkable reorientation of the virtual promastigote such that the distal tip of the flagellum is promo... |
6(b) ), where we see a rapid change in angle [MATH] when in close proximity to the boundary. This may again be partially explained by the general tendency of pullers to align perpendicular to a wall, but the large magnitude of the effect suggests that it is also resultant of the same drag-based mechanism as the beat pl... |
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