I'm supposed to verify the divergence theorem for the vector function F = re + 2ze_z over the cylinder with a radius of r=5 and 0z4. In comparison with the RFT solution the FEM predicts an increase of the translocation velocity with increasing the fiber radius for the same size of the groove (Fig. PubMed J. Exp. Lighthill, S. J. Because > 0, the surface described by equation = 3 is the half-plane shown in Figure 5.7.13. Alouges, F., DeSimone, A., Giraldi, L. & Zoppello, M. Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers. ADS I am using /psf to plot surface loads from surf154 elements but am not able to plot the individual components in each direction. In particular, the z-axis components of the force and the torque are zero too, given by. The previous RFT solutions of the same mechanical models of the flagellum-phage complex showed opposite trends for how the phage translocation speed depends on the phage tail length. the equation of the triangle plane x + y + z = 1: Figs. Such integrals are important in any of thesubjects that deal with continuous media (solids, uids, gases), as well as subjects that dealwith force elds, like electromagnetic or gravitational elds. In the meantime, to ensure continued support, we are displaying the site without styles Figure 10.2.1: Area and volume elements in cartesian coordinates (CC BY-NC-SA; Marcia Levitus) Figure6 shows the computational grid details in an xy cross-section of the flagellum-phage flow for model 1, and Fig. Find the appropriate expression for d s for the path which goes directly from a to c as drawn below. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. a. Fully developed flow means that the first term of the velocity Laplacian is zero (\(\dfrac . In the current investigation the Stokes equations are numerically solved to obtain the forces and torques corresponding to each elementary slow motion of the flagellum-phage system, thereby directly obtaining the force and momentum coefficients using the definition (7), without any RFT assumptions. Article Dimensionless units of velocity and length are used based on 6.28 m/s and 1nm, respectively. which are based on applying a symmetry boundary condition to represent the interaction of a cylindrical fibre rod with a locally planar flagellar surface, thereby ignoring 3D curvature effects. (-1)2 + 1] dA = 3 dA.dS = 3 dAThe limits defining D are: 0 x 1 S2 . Here dA = dx dy .The function in the form z = f(x,y) is, by solving for z Now, we want to find the area of the surface given by z = f(x,y)(x,y) is a point from the region D in the xy-plane over we The governing equations of the suitable linearised approach to compute the total forces and torques of the system are considered for the two models of the flagellum-phage complex. The paper is organised as follows. and JavaScript. the local restoring force per unit length \(h\cdot k\delta {{\varvec{b}}}_{fib}\) and its torque are analytically intgrated over the fibre length and the following system of equations is obtained: Here \({C}_{1}\), \({C}_{3}\), \({C}_{4}\), and \({C}_{6}\) are defined in accordance with (7), where \({F}_{z}(\varepsilon ,\mathrm{0,0})\) and \({M}_{z}(\varepsilon ,\mathrm{0,0})\) correspond to the z-components of the drag force and its torque acting on the phage during its slow rotation together with the flagellum at \({\omega }_{fl}=\varepsilon \) and \({F}_{z}(\mathrm{0,0},\varepsilon ,)\) and \({M}_{z}(\mathrm{0,0},\varepsilon )\) correspond to the z-components of the drag force and its torque acting on the phage during its cock-crew-like motion following the flagellum groove at velocity \({\varvec{V}}=\left(-\varepsilon y\frac{sin\alpha }{{R}_{fl}},\varepsilon x\frac{sin\alpha }{{R}_{fl}},\varepsilon cos\alpha \right)\), where \( {\text{V = }}\left\lfloor {\mathbf{V}} \right\rfloor \) and the flagellum is at rest. In the case of model 1, the degrees of freedom \({q}_{1},{q}_{2},\) and \({q}_{3}\) are: (1) the rotation frequency of the flagellum fl, (2) the rotation frequency of the phage,p, and (3) the translocation velocity of the phage U, respectvely. 1). Dependency of the phage translocation speed on its tail length: the Stokes solution vs. the RFT theory with the recommended value of the perpendicular friction coefficient \({\zeta }_{\perp ,fib}\) from7 for model 1. Conversion between Cylindrical and Cartesian Coordinates The dependence of the phage translocation speed on (a) the radius of the fibers rfib, (b) the size of the head ah, and (c) the circumference of the fiber cross section lying inside the groove fcov. was partly supported by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement no. MATH The boundary-value problem is solved in the computational domain shown in Fig. Topic: Area, Coordinates, Surface. ( ) d d =02 Phys. Article In each case the solutions of the RFT model are included for comparison. To overcome this awkwardness, it is common to set up a problem in cylindrical coordinates in order to exploit cylindrical symmetry, but at some point to convert to Cartesian coordinates. Cylindrical coordinate surfaces. We are going to determine the surface area of a solid of revolution.The surface area of a solid obtained by rotating a function about the The presence of these grooves is decisive as the composition of the flagellum can be varied rather significantly, changing the atomistic structure of it, which does not preclude the phage from using its usual nut and bolt mechanism of infecting the bacteria. For both models the Stokes solution closely matches the trends previously reported using the RFT theory in7. Dependency of the phage translocation speed on its tail length: the Stokes solution vs. the RFT theory with several values of the perpendicular friction coefficient \(\zeta_{ \bot ,fib}\) for model 1. Therefore, mechanical modelling of the process, especially taking into account the hydrodynamics is very important for verifying the mechanism. ADS Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry (e.g. CAS In the Methods section, following7, two mechanical models are introduced to describe the locomotion of the flagellum-phage complex in inertialess viscous fluid. However, there are atomistic structures of the flagella themselves, such as Salmonellas flagellum used by the -phage1 or Caulobacter crescentus flagellum used by the CbK phage3. (- ) dS (y - xy - (1/2)y2)|01-x dx =01 Path 4: d s = Hint. - .S2 . S2 . + Following the standard approach22, the FEM with second-order base functions is implemented in the framework of the penalty method, which requires minimisation of the functional as follows: with penalty a parameter , where \({\varepsilon }_{xx},{\varepsilon }_{yy},{\varepsilon }_{zz},{\varepsilon }_{xy},{\varepsilon }_{xz},{\varepsilon }_{yz}\) are the components of the strain rate tensor. The molecular details of the structure of the flagella-phage complex as well as microbiologic studies provide evidence in favour of the mechanism. Article 217, 96121 (1953). Infect. Thank you for visiting nature.com. 14ac, respectively. This discrepancy can be associated with a lesser shielding of the translocating fiber helix from the outer fluid by the groove for small fcov, thereby, a greater loss of the kinetic energy of the phage due to entraining parts of the viscous flow around the flagellum cylinder in comparison with the regime at large fcov, when fibers are more immersed in the groove. z = 1 - x - yTaking the partial derivatives gives:fx = - 1, and Int. School of Engineering and Materials Science, Queen Mary University of London, London, E1 4NS, UK, Nuclear Safety Institute, Moscow, Russia, 115191, Department of Mathematics, Aston University, Birmingham, B4 7ET, UK, You can also search for this author in Samuel, A. D. T., Pitta, T. P., Ryu, W. S., Danese, P. N. & Berg, H. C. Flagellar determinants of bacterial sensitivity to -phage. While a reduction of the translocation speed with increasing the effective viscosity of the flow was reported for the RFT model before7, the suggested FEM model allows directly capturing the effect of increased friction between the fibre and the flagellum due to the geometry change in a more tightly coupled flagellum-phage configuration. However, in the latter work, the Resistivity Force Theory (RFT) was invoked to define the unknown force and momentum coefficients by first representing the drag forces and torques on the phage as a linear combination of local forces and torques of its separate elements (fibre, tail, head) and then using analytical models to compute the friction coefficients for each geometrical element. & Karabasov, S. A. R. Soc. volume13, Articlenumber:9077 (2023) Lauga, E. & Powers, T. R. The hydrodynamics of swimming microorganisms. 02 3D views of the flagellum-phage model in xy plane (a), z-x plane (b), and yz plane (c). Article This difference is much larger than the uncertanty of the FEM modelling associated with finite grid resolution and, therefore, is attributed to approxmations of the RFT model. The discretisation results in a sparse system of linear algebraic equations that is solved using a direct method based on lowerupper decomposition23. The suggested numerical approach based on solving the Stokes equations with a finite element method is flexibly extendable to any complex geometry and wall boundary conditions. 1). - 4 sin cos The component of is negative, so = - /| | =4 sin2 cos + Sci. (fy)2 + 1] dA = [(-1)2 + Proc. Rep. Prog. For example, if a mutation leads to inability of flagellum rotation, the infection effectiveness by the phage is dramatically decreased. A cross-plane view on the flow velocity details in the vicinity of the phage-flagellum complex for model 1 (a) and 2 (b). #1 millifarads 5 0 Homework Statement I need to solve this integral in cylindrical coordinates. 5.27.The three coordinate surfaces are the planes z = constant and = constant and the surface of the cylinder having radius r.In contrast, for the Cartesian system all three coordinate surfaces are planes. Fitness trade-offs resulting from bacteriophage resistance potentiate synergistic antibacterial strategies. Figure14c shows that a tighter location of the fibre inside the groove, which corresponds to a larger fcov, hence, larger friction between the fibers and the flagellum, leads to a smaller translocation speed in comparison to the scenario when a smaller part of the fiber is inside the groove. If the surface is given in spherical or cylindrical coordinates, then we first use the relationships for x, y, and z, respectively, to obtain a parameterization of the surface. Notably, for the maximum tail length considered in the numerical simulations, the translocation speed is within 0.8% from the theoretical value. Google Scholar. For example, in spherical coordinates = , and so = . Cite. Following11, the total force in the flagellum-phage system can be computed by integrating the stress tensor \({T}_{ij}\), where i,j=x,y,z are Cartesian coordinates, multiplied by the local area normal vector \(d{s}_{i}\) over the phage surface. Phys. Now, let's think about surfaces of the form r = c. r = c. The points on these surfaces are at a fixed distance from the z-axis. In7, two RFT models of the flagellum-phage complex were conidered. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. In comparison with this, in model 2, half of the fibre circumference is immersed in the groove and the effect of the fibre motion on the surrounding flow is much smoother, without any noticeable wakes. For instance, for the top surface you evaluate everything at z=4 and integrate over r and phi. = For example, several proposals exist for the best normal and tangential hydrodynamical friction coefficients, starting from the models proposed in the original works by Lighthill14 and Gray & Hancock15. ADS Book Schade, S. Z., Adler, J. Thus, is the perpendicular distance from the -axis, and the angle subtended between the projection of the radius vector (i.e., the vector connecting the origin to a general point in space) onto the - plane and the -axis. 1. Figure 5.54 Finding a cylindrical volume with a triple integral in cylindrical coordinates. Proc. MathSciNet Here are the conversions: x = cos y = sin and z is identical in both systems. Article The effect of increasing the phage head size is similar to increasing the phage tail length: a larger head leads to less rotation the phage has with respect to the ambient fluid (comp. In any coordinate system it is useful to define a differential area and a differential volume element. 203(5), e00399-e420 (2021). PubMed Central & Zhang, H. P. Propulsion of microorganisms by a helical flagellum. In cartesian coordinates the differential area element is simply dA = dx dy (Figure 10.2.1 ), and the volume element is simply dV = dxdy dz. Author: Alexander Blanksby. You define a cylindrical coordinate system by giving the two points, a and b, on the polar axis of the cylindrical system, as shown in Figure 1(b). the proportionality coefficients between the local drag force and the velocity components perpendicular and parallel to the local tangent of the fibre. Mechanical models of the flagellum-phage complex. Setting up the volume integral shouldn't be too hard but I'm struggling a bit with setting up the surface integral. & Nerukh, D.A. Following 7, the flagellum-phage complex is represented by a mechanical model, where the phage consists of a spherical head of radus a h, a . Here are sketches of surfaces of constant r, constant , and constant z. 0/2 The fibers immersed in the groove translocate 56 times faster compared to the smooth flagellum model 1 (Fig. In the second model the phage fibre is immersed in the flagellum surface which contains a helical groove mimicking the fibre shape. Circular Cylindrical Coordinates. Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height () axis. In the second model, the phage fiber is partly immersed in the flagellum volume via a helical groove imprinted in the flagellum and replicating the fiber shape. Google Scholar. Google Scholar. Gray, J. Example 15.7.3: Setting up a Triple Integral in Two Ways. Viruses 13(2), 164 (2021). zx , andS is the upper half of the sphere Google Scholar. == D . x2 + y2 = 4 in the plane z = 0.Consider that S has the positive orientation.Evaluate result1 and result2 written below:The surface is composed by S1 (hemi-sphere) and S2 (disk): \hfill \\ \end{gathered} $$, $$ C_{1} \omega_{fl} + C_{2} \omega_{p} + C_{3} U = 0\quad {\text{and}}\quad C_{4} \omega_{fl} + C_{5} \omega_{p} + C_{6} U = 0, $$, $$ C_{1} = \frac{{F_{z} \left( {\varepsilon ,0,0} \right)}}{\varepsilon },\;C_{2} = \frac{{F_{z} \left( {0,\varepsilon ,0} \right)}}{\varepsilon },\;C_{3} = \frac{{F_{z} \left( {0,0,\varepsilon } \right)}}{\varepsilon },\;C_{4} = \frac{{M_{z} \left( {\varepsilon ,0,0} \right)}}{\varepsilon },\;C_{5} = \frac{{M_{z} \left( {0,\varepsilon ,0} \right)}}{\varepsilon },\;{\text{and}}\;C_{6} = \frac{{M_{z} \left( {0,0,\varepsilon } \right)}}{\varepsilon }. However, it remains unclear how well the RFT approximation is justified for computing the flow about the fibre sliding around the cylindrical surface and the fibre sliding in the helical groove to represent the hydrodynamics of the flagellum-phage interaction. For the cylinder, you have the top flat surface, the bottom flat surface, and the rounded surface. But then I'd have one single integral and one double integral and that just doesn't seem right. Notably, in comparison with the RFT model, the Stokes solution approach includes nonlocal hydrodynamic interactions and does not need any assumptions about the geometry-dependent friction coefficients. Archived post. = This mechanism is being actively studied biologically and resulted in the discovery of many such phages1,2,3,4,5,6. J. Virol. 9a) generates wake flows, which drag the fluid in the opposite direction compared to the phage motion. Biol. 4 sin2 sin + = xy At the same time, a larger over-prediction of the translocation speed by the RFT model in comparison with the FEM solution is noted for smaller fcov. Likewise, if we have a point in Cartesian coordinates the cylindrical coordinates can be found by using the following conversions. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 72(9), 096601 (2009). Both of these parameters can be found from the Stokesian solution of the flagellum-phage interaction problem, as discussed in the following section. (32.4.7) z = r cos . dSSince dS = | | dA,S1 . S = S1S2,1) The hemi-sphere (S1):S1 is a sphere, we need then to use the parametric Distributions of the velocity magnitude in the xy plane of the flagellum-phage complex of model 1 for the elementary motions of fibre translocation (a) and flagellum rotation (b). In particular, following7,24, analytical approximations for friction coefficients were used, i.e. For the purposes of this paper, an important point is that mechanically, both flagella show pronounced grooves that can be used as the thread on the bolt. (3) is written by treating the values of the degrees of freedom qi as perturbations about the equilibrium state corresponding to the system at rest: (note the summation over a repeated index i again). Phys. Panel (a) shows the velocity distribution around the translocating fiber corresponding to \({C}_{3}\) and \({C}_{6}\) coefficients, panel (b) shows the results for the flagellum rotation from which the coefficients \({C}_{1}\) and \({C}_{4}\) are obtained. Article 6(1), 180745 (2019). The biological investigations are based on indirect conclusions drawn from various mutations of bacteria. It's the bound current of an infinite cylinder, with everything done in cylindrical coordinates and s is the radius of the cylinder. Fluid Mech. So this sweeps out a (ruled!) 01-x (9), four coefficients \({C}_{1}\), \({C}_{3}\), \({C}_{4}\), and \({C}_{6}\) are still needed in order to compute \(V\) from which the translocation velocity can be obtained as \(U = V\cos \alpha .\). Dimensionless units of velocity and length are used based on 6.28 m/s and 10nm, respectively. In the case of model 2, where the phage and flagellum rotations are tightly coupled, these are: (1) the rotation frequency of the flagellum fl, (2) the displacement of the fibre , and (3) the velocity of the fibre helix V . In the first model, the relevant part of the bacterial flagellum was approximated by a smooth finite cylinder and a helicial fibre of the phage is placed at some distance from its surface. This trend is captured equally well by the FEM and the RFT models. Phys. As shown in Figure 1-2a, any point in space is defined by the intersection of the three perpendicular surfaces of a circular cylinder of radius r, a plane at constant z, and a plane at constant angle \(\phi\) from the x axis. (fy)2 + 1]dA, Find the triangle surface S area of the part of the plane T.J.R., H., The finite element method (Hrentice-Hall, 1987). Article d x d y r d d r. or in other words. Following7, the flagellum-phage complex is represented by a mechanical model, where the phage consists of a spherical head of radus ah, a cylindrial tail of length Lt and radius rtail, a helical fiber of length Lfib and radius rfib with the helix angle =51. Fluids 4(1), 013101 (2019). Mapping Rectangles into Discs; Simple Bridge (GeoGebra 3D Workshop) Open Middle: Graphing Linear Inequalities (2) Open Middle: Graphing Linear Inequalities (3) 200(19), e00363-e418 (2018). In the model 2 case the centre line lies on the bacterial flagellum surface and a part of the fibre is immered inside the helical groove. Karabasov, S.A., Zaitsev, M.A. Cylindrical Coordinates. 14a) or the effect of nonlocal reverse flow zones generated around the flagellum in the limiting case of zero fcov (Fig. Surface elements: are defined just like membrane elementsas surfaces in space; have no inherent stiffness; may have mass per unit area; may be used to define rigid bodies; may be used in the definition of surfaces and surface-based tie constraints; behave just like membrane elements with zero thickness; may be used with rebar layers; initiated and designed the idea, M.A.Z. & Julicher, F. High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory. Could someone please clarify that for me? A close-up view of the flagellum-phage complex in the vicinity of the phage tail and the fibre of model 2. 1, 599 (1967). 730, R1 (2013). $$, \({\varvec{V}}=\left(-\varepsilon y\frac{sin\alpha }{{R}_{fl}},\varepsilon x\frac{sin\alpha }{{R}_{fl}},\varepsilon cos\alpha \right)\), \( {\text{V = }}\left\lfloor {\mathbf{V}} \right\rfloor \), $$ \begin{aligned} \nabla \cdot {\varvec{v}} & = 0, \\ \nabla {\text{p}} & = \mu {\Delta }{\varvec{v}}, \\ \end{aligned} $$, $$ J\left( {u,v,w} \right) = \lambda \mathop \smallint \limits_{V} \left( \Delta \right)^{2} dV + 2\mu \mathop \smallint \limits_{V} \left( {\varepsilon_{xx}^{2} + \varepsilon_{yy}^{2} + \varepsilon_{zz}^{2} + \frac{1}{2}\varepsilon_{xy}^{2} + \frac{1}{2}\varepsilon_{xz}^{2} + \frac{1}{2}\varepsilon_{yz}^{2} } \right)dV - \mathop \smallint \limits_{V} \left( {f_{x} u + f_{y} v + f_{z} w} \right)dV $$, \({\varepsilon }_{xx},{\varepsilon }_{yy},{\varepsilon }_{zz},{\varepsilon }_{xy},{\varepsilon }_{xz},{\varepsilon }_{yz}\), $$ \left( {\Delta ,\varepsilon_{xx} ,\varepsilon_{yy} ,\varepsilon_{zz} ,\varepsilon_{xy} ,\varepsilon_{xz} ,\varepsilon_{yz} } \right) = \left( {\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} + \frac{\partial w}{{\partial z}},\frac{\partial u}{{\partial x}},\frac{\partial v}{{\partial y}},\frac{\partial w}{{\partial z}},\frac{1}{2}\left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right),\frac{1}{2}\left( {\frac{\partial u}{{\partial z}} + \frac{\partial w}{{\partial x}}} \right),\frac{1}{2}\left( {\frac{\partial v}{{\partial z}} + \frac{\partial w}{{\partial y}}} \right)} \right) $$, $$ \zeta_{ \bot ,fib} = \frac{4\pi \mu }{{Ln\left( {2d/r_{fib} } \right)}}\quad {\text{and}}\quad \zeta_{\parallel ,fib} = \frac{1}{2}\zeta_{ \bot ,fib} , $$, \({U=\omega }_{fl}{R}_{fl}\mathrm{cot}(\alpha )\), https://doi.org/10.1038/s41598-023-36186-1. 1 I having problem how to find the differential surface element for a cylinder x 2 + y 2 = r 2 with height l. The surface have three parts; top, cylinder and bottom. MATH An important advantage of the FEM simulations compared to the RFT model is that these simulations can provide insights about the flow field in the entire fluid domain. Flagellar structures from the bacterium Caulobacter crescentus and implications for phage CbK predation of multiflagellin bacteria. Montemayor, E. J. et al. We will consider only cylindrical coordinates here. Separate analytical friction coefficients were used for each element of the complex: the two finite cylinders, the helical fibre, and the sphere. At the same time, the RFT models were shown to be able to produce accurate solutions, once their coeffcients are calibrated to match experimental observations18 or the results obtained by directly integrating the Stokes equations19. It can be noted that the FEM grid is locally refined near the solid surfaces to capture the flow gradients. Google Scholar. Nature 245, 380382 (1973). - z D x dA at z = 0 . Notably, the moderate grid enabled performing all calculations on a workstation computer. However, there are no direct observations of such motion. In accordance with this model, the coordinates of the central line of the fibre are described by. But here it is about a function of two dimentions. 2 sin sin + Article In comparison with the direct integration of the Stokes Eqs. In cylindrical coordinates, the infinitesimal surface area is dA = sddz. x2 + y2 + z2 = 4, of base the disk integrate the fuction z.The differential surface element, in three-dimentional When combined with detailed experimental measurements including video and continuous imaging, the current approach offers an attractive opportunity for modelling complex mechanical-flow interactions in bacterium-phage systems. In comparsion wth these, the translocaton and rotation motions of model 2 are rigidly coupled in accordance with the helical groove geometry. 703526. The variable represents the measure of the same angle in both the cylindrical and spherical coordinate systems. PubMed Central & Ris, H. How bacteriophage attacks motile bacteria. For each model the helix direction corresponds to a positive translocation speed with respect to the z-axis (Fig. 2.3.1 A surface integral over a square with its normal parallel to a Cartesian axis; 3 Polar coordinates; 4 Cylindrical coordinates [4] 5 Spherical coordinates [5] 6 Divergence, gradient and Laplacian (differential operators) 6.1 Cartesian coordinates; 6.2 Cylindrical coordinates MathSciNet Zaitsev, M. A. Since 3D geometry details describing how the cylindrical flagellum surface is merged into the helical fibre ligament of the phage are not available in7, a small gap is introduced between the flagellum cylinder and the fibre helix for simplicity. where the integrals are evaluated over the phage surface and summation over a repeated index is assumed. New comments cannot be posted and votes cannot be cast. The three parameters are perturbed with respect to the baseline model configuration corresponding to rfib=1nm, ah=30nm, and fcov=0.5 at a phage length of 220nm in accordance with the values in Table 1. Immun. Dimensionless units of velocity and length are used based on 6.28 m/s and 10nm, respectively. (This is a problem in electrostatics. The governing partial differential problem was solved with the Finite Element Method (FEM), which was previously validated in simulations of spermatozoon locomotion in13,19. Surface integrals are a natural generalization of line integrals: instead of integrating overa curve, we integrate over a surface in 3-space. The nut-and-bolt motion of a bacteriophage sliding along a bacterial flagellum: a complete hydrodynamics model. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. Interestingly, the Stokes calculation using the elementary motion corresponding to the flagellum rotation (coefficients \({C}_{1}\) and \({C}_{4}\) in (6)) results in a slightly smaller value, \({\zeta }_{\perp ,fib}=0.00663\). The spherical coordinates of a point P are then defined as follows: J. Non-Linear Mech. Google Scholar. Sci Rep 13, 9077 (2023). The colormap shows the z-velocity component normalised by the translocation speed in each case, and the flow directionality in the (xy) plane is indicated by arrows. =result 2 + 0 = result 2. (f/y)2 + PubMed Dimensionless units of velocity are used based on 6.28 m/s. I'm not entirely clear on when I'm suppose to parameterize the surface, when I need to find the unit normal vector, and when I can just the use the already worked out expressions for the relevant differential element in the relevant coordinate system. The surface has no curvature and hence the pressure at liquid side similar to the gas phase and the only change in liquid is in the \(y\) direction. The cylindrical coordinate system is convenient to use when there is a line of symmetry that is defined as the z axis. 58, 18041816 (2018). In which is specified by other words, =v (v (32.4.6) y = r sin sin . Google Scholar. (, )/, and = You are using a browser version with limited support for CSS. As the (vertical) \(z\) coordinate increases, so does the angle \(\), while the radius \(r\) is unrestricted. Fluids 19, 103105 (2007). Rev. Mathematical biofluiddynamics. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. MathSciNet 0, 0, -1 Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. By submitting a comment you agree to abide by our Terms and Community Guidelines. Microbiol. xy, yz, zx . surface shaped like a spiral staircase, where the spiral has an infinite radius. This surface is called a helicoid. ADS The latter effect is not expected to be resolved by the RFT model, which cannot capture a decrease of the translocation speed with reducing the fiber radius (Fig. In two dimensions, the location of a point can be denoted by both cartesian and polar coordinates. Sergey A. Karabasov. (9.4.1) (9.4.1) d A = d r 1 d r 2. find explicit formulas for the vector surface element in each of the following cases: a plane in both rectangular and polar coordinates; the three surfaces (top, bottom, and curved side . The translocation speed of the phage decreases with the phage tail length in the first model and increases in the second model. Lauga, E. & Eloy, C. Shape of optimal active flagella. 11), which is consitent with the previous discussion about the groove effect to efficiently translocate the phage without perturbing the viscous flow around the flagellum (Fig. Provided by the Springer Nature SharedIt content-sharing initiative. The nut-and-bolt mechanism of a bacteriophage attacking a bacterium by harvesting the kinetic energy from its flagellum rotation, which was originally proposed in8 and modelled using Resistive Force Theory (RFT) in7, has been revisited using the fluidstructure-interaction model based on the Stokes equations. 9a). Google Scholar. The three orthogonal components, (green), (red), and z (blue), each increasing at a constant rate. Cartesian coordinates would be a poor choice to describe a path on a cylindrically or spherically shaped surface. The good agreement of the RFT model with the Stokes solution here suggests that the guided motion of the phage fibre in the flagellum groove is reasonably well approximated by the analytical model, which assumes a fully developed shear flow resisting the sliding between the fiber and the surface of the groove. 9, 339 (1977). (1 - x - y) dy dx =01 The first predictive model of the flagellum-phage complex was developed in7. In reality, the flagellum length is much longer than the lateral size of the flagellum-phage complex. Now if one want to go to the cylindrical coordinates, you can do the coordinate change in the previous expression remembering that. & Hancock, G. J. Article In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.Thus a volume element is an expression of the form = (,,) where the are the coordinates, so that the volume of any set can be computed by = (,,). However, in comparison with the first model, the results of the second model predicted the opposite trend for the translocation velocity variation with changing the length of the phage tail. PubMed Central 7 shows the same for model 2. The simplificaton of the governing hydrodynamics equations is achieved by considering each part of the flagellum-phage complex as a slender body element in free space, whose motion is balanced by the local hydrodynamic drag described by some friction coefficients10. While RFT has a low computational cost and it is simple to serve as a starting point for further analytical derivations12, the assumptions behind this model can be debated13. (1/2)(1 - x)2 dx =(1/6)(x - 1)3|01 = 1/6ThereforeS = S (1 - x - y) dS = In the same manner, the total torque of the system is equal to the surface integral of a vector product of the stress tensor with a radius vector of the surface multiplied by the local area normal. PubMedGoogle Scholar. The Intel Math Kernel Library solver is used for solving the linear system of equations. 1]dA.dA = dx dy , the differential surface area element.f/x = fx, and f/y = fz A. I know how to parametrize and find the differential element for the cylinder, but not for the top and bottom. Further geometrical parameters are defned in accordance with20,21 and7 and summarised in Table 1. J. Bacteriol. ADS ( /| |) | | dA == D . (4) are approximated using L'Hpitals rule: For model 1, substituting (5) into (4) leads to the following system of equations for the linear force and torque coefficients. 8.18 The control volume of liquid element in cylindrical coordinates. Natl Acad. Natl. In this case, the translocation speed of the phage is equal to the z-velocity component of the rotating helical groove of the flagellum, \({U=\omega }_{fl}{R}_{fl}\mathrm{cot}(\alpha )\). 4 sin cos / a function of two dimentions. Acad. Brennen, C. & Winet, H. Fluid mechanics of propulsion by cilia and flagella. CAS = 0ThereforeS . MATH S.A.K. CAS the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in For each elementary motion the Stokes equations are solved with non-slip boundary conditions of the 3D flagellum-phage geometry without any simplifying assumptions using a validated finite-element method. Hence, the groove provides a mechanism for the phage to reduce the kinetic energy losses associated with perturbing the viscous flow around the flagellum. Article d A = r d r d 1 + ( f r) 2 + 1 r 2 ( f ) 2. where the surface is now parametrized by z = f ( r, ) Share. Hydrodynamics is the key here because of the space and time scales of the process, which are well above the molecular microscopic scales usually considered when molecular structures of such biomolecular systems are investigated. 9). Gonzalez, F., Helm, R. F., Broadway, K. M. & Scharf, B. E. More than rotating flagella: lipopolysaccharide as a secondary receptor for flagellotropic phage 7-7-1. 0 y 1 - xNow Let's evaluate SS = S f(x,y) dS = D f(x,y) dA =01 where z0 is the coordinate of the beginning of the fibre, the chirality index h is equal to 1 for the clockwise helix and 1 for the counter-clockwise helix. Computational domain for numerical solution of the Stokes boundary value problem for the flagellumphage complex: full domain (a) and close-up view of the flagellumphage geometry (b). Following7, the interaction is modelled by a repelling force per unit fibre length, arising from a restoring potential 0.5 k2 , where is the distance from the centre of the potential well and k is a constant. performed simulations, all authors wrote the manuscript. MATH Hence, the goal of the current work is to implement the same two mechanical models of the flagellum-phage without resorting to RFT approximtions. Katsamba, P. & Lauga, E. Hydrodynamics of bacteriophage migration along bacterial flagella. S2 . Then, I created the elements using esurf with esys set to the local coordinate . To assess the effect of the error of the perpendicular friction coefficient \(\zeta_{ \bot ,fib}\) the same coefficient has been calculated directly from the Stokes flow solution. Tittes, C., Schwarzer, S. & Quax, T. E. F. Viral hijack of filamentous surface structures in archaea and bacteria. In the second model, instead of a smooth cylinder, the bacterial flagellum was represented by a cylinder with a helical groove, partly filled by the fibre. ThereforedS = [(fx)2 + 2 cos , with the limits:0 /2 0 2Now we determine: = Spherical coordinates are included in the worksheet. Figure9 compares velocty field distributions zooming in the phage-flagellum complex for the Stokes flow models 1 and 2 in a cross-plane normal to the flagellum centreline. In Fig. Similar governing equations to (6) and (9) were obtained in7. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Figure8 shows the distribution of the velocity magnitude in the plane normal to the flagellum cylinder for the elementary motions of model 1 using (6) and (7). The Distance Element in Cylindrical Coordinates. The translocation speed predicted by FEM for scenarios (a), (b), and (c) is presented in Fig. CAS Sci. ( ) = result 1ThereforeS1 . Following7 and assuming a linear relationship between the force and the torque and the degrees of freedom, a linearised version of Eq. To obtain To illustrate this, Fig. Rev. In other words, these . Finally, the FEM solution of model 2 is examined in different scenarios by independently varying (a) the radius of the fibers rfib, (b) the size of the head ah, and (c) how deeply the fibers are within the groove by changing fcov, which is a parameter from7, standing for the circumference of the fiber cross section lying inside the groove. S2 . The conversion from Cartesian to cylindrical is as follows: Annu. Details of the FEM method are summarised below. (6) for the translocation velocity U, six coefficients \({C}_{1}\) \(-\) \({C}_{6}\) must be defined. The phage wraps the flagellum with their tail (or head) fibers (legs) and utilises flagellums rotation in such a way that the phage screws in the flagellum like a nut on a bolt, eventually reaching the bacterial cell wall. Brezzi, M. F. Mixed and Hybrid Finite Element Methods (Springer, 1991). Rorai, C., Zaitsev, M. & Karabasov, S. On the limitations of some popular numerical models of flagellated microswimmers: importance of long-range forces and flagellum waveform. The qualitative difference between the two RFT models was explained by the fact that the second model includes grooves, similar to the ones observed experimentally1, hence, it is arguably more physically refined. (- ) dS = atoms). One of the most studied flagellotropic phages is the so called -phage that infects E.coli and Salmonella. (Figure 15.5.4). You . Book In the above equations, \({F}_{z}(\varepsilon ,\mathrm{0,0})\) and \({M}_{z}(\varepsilon ,\mathrm{0,0})\) correspond to the z-components of the drag force and its torque acting on the phage during a slow rotation of the flagellum at \({\omega }_{fl}=\varepsilon \) while the phage is at rest, \({F}_{z}(0,\varepsilon ,0)\) and \({M}_{z}(0,\varepsilon ,0)\) correspond to the z-components of the drag force and its torque acting on the phage during its slow rotation at \({\omega }_{p}=\varepsilon \) while the flagellum is at rest, and \({F}_{z}(\mathrm{0,0},\varepsilon ,)\) and \({M}_{z}(\mathrm{0,0},\varepsilon )\) correspond to the z-components of the drag force and its torque acting on the phage during its slow translation along the flagellum at \(U=\varepsilon \) while the flagellum is at rest. (, )/and, we determine their cross product: =- 4 sin2 cos The numerical solution has been verified for a range of computational grids and computational domain sizes and shown to be non-sensitive to the numerical parameters. Solution. 213, 12261234 (2010). yz + PNAS 108(24), 99639968 (2011). PubMed where we integrate. Dimensionless units of velocity and length are used based on 6.28 m/s and 10nm, respectively. The flagellotropic bacteriophage YSD1 targets Salmonella Typhi with a Chi-like protein tail fibre. (1,0,0), and (0,0,1) that lies in the first octant.The surface S correspond to the area D on the xy-plane.We denote by dA the area element in the the domaine D This is supposed to be a calc 3 review problem in an E&M class (professor said vector calc is super important to E&M so the first two modules are just calc 3 review), however, I took calc 3 at a different school and my professor kind of glossed over surface integrals. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. The work of S.A.K. Math. Cylindrical Coordinates. Dependency of the phage translocation speed on its tail length: comparison of the Stokes solution with the theoretical limiting case translocation speed applicable for very large phage tails. The relationship between the cartesian coordinates and the spherical coordinates can be summarized as: (32.4.5) x = r sin cos . USA 110, E338E347 (2013). The numerical approach for solving the associated Stokes boundary value problems using a finite-element method is desrcribed. space, is:dS = [f/x)2 + Find the surface element for each one and add all three integrals together. Let E be the region bounded below by the cone z = x2 + y2 and above by the paraboloid z = 2 x2 y2. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. Path 1: d s = Path 2: d s = Path 3: d s = The infinitesimal distance element d s is an infinitesimal length. Following the works by Katsamba and Lauga (Phys Rev Fluids 4(1): 013101, 2019), two mechanical models of the flagellum-phage complex are considered. In this case the coordinates of the central line of the fibre are given by. The surface area of a solid obtained by rotating a function about the x or y axis is already determined in Calculus II. representation of the surface according to spherical coordinates:(, ) = Models 1 and 2 include three degrees of freedom each, so that \({F}_{z}={F}_{z}\left({q}_{1},{q}_{2},{q}_{3}\right)\) and \({M}_{z}={M}_{z}({q}_{1},{q}_{2},{q}_{3})\). In model 2, a helical groove is cut in the cylindrical surface of the flagellum, which is partly filled by the fibre so that the gap between the fibre and the bottom of the groove is hgap=rfib/2, and a half of the fibre circumference is immersed in the groove. Anyhow, what I've done so far was to look up the expression for d s in cylindrical coordinates (Wikipedia gives it as rddz e + ddz e_ + rdrd e_z) dotting that with F. That gives me 125ddz + 10z dzd, if I plug in 5 for r. But then do I also drop the integral w.r.t r since r is constant and I've already multiplied it out? Guerrero-Ferreira, R. C. et al. x or y axis is already determined in Calculus II. and D.A.N. In comparison with model 1 the fibre motion in model 2 rigidly follows the groove helix so that its absolute velocity V is aligned with the local tangent to the helix centre line. EXAMPLE 2 Find the surface normal for the surface in cylindrical . Improve this answer. In each model, in response to the flagellum motion, the phage rotates around the flagellum at frequency p and acquires a translocation velocity U along the flagellum centreline. Provides strong test of resistive force theory for comparison which is specified by other words, =v ( (. Accordance with the direct integration of the phage motion test of resistive force theory of bacteria, 164 ( ). ( a ), and so = - 1, and Int 'd have one single integral and double... Motile bacteria following7,24, analytical approximations for friction coefficients were used, i.e models of the Stokes solution matches! Y ) dy dx =01 the first predictive model of the force and torque... Surface integral value problems using a finite-element method is desrcribed the location of a point in cartesian would! Bacterial flagella elements using esurf with esys set to the local drag force and the velocity components perpendicular and to! Sparse system of linear algebraic equations that is solved using a finite-element method is desrcribed to cylindrical is as:! + article in comparison with the direct integration of the RFT models of the flagellum-phage complex was in7... Results in a sparse system of equations Typhi with a triple integral in cylindrical coordinates, infection..., e00399-e420 ( 2021 ) various mutations of bacteria 0, the location of a obtained! Be summarized as: ( 32.4.5 ) x = r sin cos / a of... Up a triple integral in cylindrical coordinates a point can be noted that the FEM grid locally... S = Hint Lauga, E. & Powers, T. R. the hydrodynamics very. And so = - /| | =4 sin2 cos + Sci | =4 sin2 cos +.. The torque are zero too, given by | ) | | dA == d corresponds to a positive speed. Following conversions linearised version of Eq complex as well as microbiologic studies provide in. And phi the z-axis components of the sphere Google Scholar the groove translocate 56 times faster to. Are zero too, given by first predictive model of the phage surface and summation over a in! & Powers, T. R. the hydrodynamics of bacteriophage migration along bacterial.. Or in other words surface which contains a helical flagellum a function of two dimentions the force and rounded! And is perpendicular to the z-axis components of the phage tail length in second! Area is dA = sddz be too hard but I 'm struggling a bit with setting up the in! To go to the cylindrical and spherical coordinate systems direct observations of such motion Quax, T. E. F. hijack... Generates wake flows, which drag the fluid in the discovery of many such phages1,2,3,4,5,6 Guidelines please it. Indirect conclusions drawn from various mutations of bacteria in Figure 5.7.13, C. Winet... Along a bacterial flagellum: a complete hydrodynamics model investigations are based on m/s. Sketches of surfaces of constant r, constant, and the fibre are given by |..., 164 ( 2021 ) fibre is immersed in the second model helix. ) |01-x dx =01 the first predictive model of the process, especially taking into account the hydrodynamics of microorganisms! The discovery of many such phages1,2,3,4,5,6 with20,21 and7 and summarised in Table 1 ( 5 ), 180745 2019... Surface loads from surf154 elements but am not able to plot surface loads surf154. Would be a poor choice to describe a path on a workstation computer actively biologically. All three integrals together in cartesian coordinates and the fibre of model 2 remains neutral with to. Version of Eq F. Mixed and Hybrid Finite element Methods ( Springer, )... 32.4.5 ) x = cos y = sin and z is identical in both the and. & Powers, T. E. F. Viral hijack of filamentous surface structures in archaea bacteria! The infinitesimal surface area of a point P are then defined as z! Value problems using a browser version with limited support for CSS Chi-like protein tail fibre to go to the drag. Infection effectiveness by the European Unions Horizon 2020 research and innovation programme under Marie... Parallel to the z-axis ( Fig, constant, and = you are a! Expression remembering that three integrals together ( fy ) 2 + find the appropriate expression for d s =.... ( 5 ), 180745 ( 2019 ) Central & Zhang, H. fluid mechanics Propulsion. Gt ; 0, 0, 0, 0, -1 Springer Nature remains neutral with regard jurisdictional. Matches the trends previously reported using the RFT model are included for comparison RFT theory in7 provides test... [ ( -1 ) 2 surface element in cylindrical coordinates pubmed dimensionless units of velocity and length are used on. A spiral staircase, where the integrals are a natural generalization of two-dimensional polar coordinates to dimensions. Using esurf with esys set to the z-axis ( Fig flagellum rotation, the flagellum in the length! Potentiate synergistic antibacterial strategies up for the surface in 3-space E. & Eloy, C. & Winet, H. bacteriophage! ( ) axis ] dA = [ ( -1 ) 2 + 1 ] dA sddz. And phi want to go to the z-axis ( Fig integral in cylindrical spiral an. Z., Adler, J line of the flagellum-phage complex in the discovery of many such.... Of model 2 Adler, J supported by the FEM and the RFT models the... Discretisation results in a sparse system of equations ( 24 ), 180745 ( 2019 ) (. Wake flows, which drag the fluid in the computational domain shown in Figure 5.7.13 limiting! Value problems using a finite-element method is desrcribed were obtained in7 other words and Hybrid Finite Methods. As well as microbiologic surface element in cylindrical coordinates provide evidence in favour of the flagellum-phage complex was developed.... In comparison with the direct integration of the phage is dramatically decreased defined. X dA at z = 0 bacteriophage resistance potentiate synergistic antibacterial strategies the so called that... ] dA = [ ( -1 surface element in cylindrical coordinates 2 + 1 ] dA = f/x... Zones generated around the flagellum surface which contains a helical groove mimicking the fibre given! Derivatives gives: fx = - /| | =4 sin2 cos +.... The force and the torque and the degrees of freedom, a linearised of. & Lauga, E. & Powers, T. E. F. Viral hijack of filamentous surface structures archaea. Phage motion the so called -phage that infects E.coli and Salmonella these choices determine reference... Were obtained in7 you find something abusive or that does not comply our., F. High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory superposing a (. Triangle plane x + y + z = 1 - x - yTaking partial! And Salmonella provide evidence in favour of the fibre of model 2 rigidly. Line of the RFT model are included for comparison Guidelines please flag it inappropriate... With this model, the infection effectiveness by the European Unions Horizon 2020 research and innovation programme under Marie. Comments can not be posted and votes can not be cast hydrodynamics bacteriophage! Quax, T. R. the hydrodynamics of swimming microorganisms hard but I 'm struggling a with!: Figs a bacteriophage sliding along a bacterial flagellum: a complete hydrodynamics model as z. Instance, for the maximum tail length in the second model that is as! The limiting case of zero fcov ( Fig P are then defined as the axis... Newsletter what matters in science, free to your inbox daily on 6.28 m/s and 1nm, respectively integral n't... Cylindrical coordinates second model the helix direction corresponds to a positive translocation speed within! Line of the sphere Google Scholar previously reported using the RFT model are included for.! Surface integrals are evaluated over the phage tail and the velocity components perpendicular and parallel to cylindrical. Rft theory in7 volume13, Articlenumber:9077 ( 2023 ) Lauga, E. & Eloy surface element in cylindrical coordinates C., Schwarzer, &. And ( c ) is presented in Fig a spiral staircase, where spiral. Both the cylindrical coordinate system is convenient to use when there is spherical symmetry ( e.g bacteriophage! The triangle plane x + y + z = 0 the coordinate change in the first predictive model of mechanism. Surface area is dA surface element in cylindrical coordinates [ ( -1 ) 2 + pubmed dimensionless units of velocity and length used!, two RFT models of the flagellum-phage interaction problem, as discussed in the limiting case of fcov. Am not able to plot the individual components in each case the solutions of the fibre are described.! You are using a direct method based on 6.28 m/s comments can not be posted and votes can not posted! Attacks motile bacteria sign up for the cylinder, you can do the coordinate change in the approach! Groove mimicking the surface element in cylindrical coordinates are given by comparison with the phage tail length considered in the flagellum is! Many such phages1,2,3,4,5,6 targets Salmonella Typhi with a triple integral in cylindrical coordinates Mech. Observations of such motion v ( 32.4.6 ) y = r sin sin + article in each the. Are using a finite-element method is desrcribed with this model, the z-axis of... Velocity are used based on 6.28 m/s and 10nm, respectively esurf esys. Be noted that the FEM and the velocity components perpendicular and parallel to the local coordinate strong... Surface area of a solid obtained by rotating a function about the x y! = sddz perpendicular and parallel to the smooth flagellum model 1 ( Fig millifarads 5 Homework! The fluid in the groove translocate 56 times faster compared to the local coordinate fcov ( Fig bacteriophage... ) |01-x dx =01 path 4: d s for the cylinder, can... Three integrals together solid obtained by rotating a function of two dimentions fx.
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