The stormy fluid dynamcis of the living cellThe stormy fluid dynamicsof the living cell Flow lines from a simulation of pronuclear migration. (Courtesy of Tamar Shinar.)

The stormy fluid dynamics of the living cellDaniel Needleman and Michael Shelley

Citation Physics Today 72 9 32 (2019) doi 101063PT34292View online httpsdoiorg101063PT34292View Table of Contents httpsphysicstodayscitationorgtocpto729Published by the American Institute of Physics

The stormy fluid dynamics of the living cell

Flow lines from a simulation of pronuclearmigration (Courtesy of Tamar Shinar)

SEPTEMBER 2019 | PHYSICS TODAY 33

Daniel Needleman (dneedleseasharvardedu) is GordonMcKay Professor of Applied Physics and professor of molecularand cellular biology at Harvard University in Cambridge Massachusetts Michael Shelley (mshelleyflatironinstituteorg)directs the Center for Computational Biology at the Flatiron Institute and is the Lilian and George Lyttle Professor of AppliedMathematics at the Courant Institute of Mathematical Sciencesat New York University where he codirects its Applied Math Laboratory

Scientists have known for centuries that some plantand amoeboid cells have cytoplasmic flow inside themas illustrated in figure 1a Modern light microscopyhas shown that such directed motions in cells are quitecommon Researchers have studied those flows usingsuch sophisticated methods as particle imaging ve-

locimetry and simulations (see figures 1b and 1c)Such flows underlie the most basic biological func-tions of cells and can be a cause an effect or both Inany case understanding them requires the study offorces and stresses that are created from activity in-side the cell itself

Daniel Needleman

and Michael Shelley

Cell biology has its beginnings in the first observations of cells throughprimitive microscopes and in the formulation of cell theory which postulates that cells are the fundamental building blocks of life Lightmicroscopes showed that the insides of cells contained complex structuressuch as nuclei spindles and chromosomes The advent of electron

microscopy in the mid 20th century brought the first truly detailed views of cell innardsImages revealed complexity at all observable scales including cell-spanning networksof polymers intricate organelles made of membranes and a variety of micron- tonanometer-sized sacs and granules such as vesicles lipid droplets and ribosomes (Fora glossary of cellular components see the Quick Study by Ned Wingreen PHYSICS TODAYSeptember 2006 page 80) Those structures are immersed in or part of the aqueouscytoplasmmdashthe cellrsquos fluidic medium

The flows inside cells are intricate and

often counterintuitive Measurements

models and simulations are helping to

unravel the complexities

C

34 PHYSICS TODAY | SEPTEMBER 2019

The self-contained character of subcellular fluid mechanicsmakes its nature fundamentally different from other areas offluid dynamicsmdashweather for example Both systems are in-credibly complex but weather is driven externally by dailyand yearly cycles of rotation and orbit whereas cells are driveninternally Whatrsquos more the basic fluid-dynamical equationsand thermodynamics of the atmosphere are known That in-formation enables one to predict its short-term behavior Bycontrast although our basic understanding of the molecularconstituents that make up cellular structures is firm biophysi-cists donrsquot yet have a sophisticated enough theoretical under-standing to fully comprehend and predict a cellrsquos dynamicsNonetheless as with weather we believe that fluid mechanicsand flows will be an important part of the story

Crowded and complicatedThe cytoplasm is not a simple liquid As shown in figure 2aeven a cellular volume of a few cubic microns is filled with an aqueous slurry of components working proteins polymermeshes and organelles of different sizes and compositions1The slurry is activemdashthat is its component parts convert a localsource of (usually chemical) energy to change their shape andgenerate forces And the slurry can change markedly over thelife of the cell For example organelles such as the endoplasmicreticulummdashessentially tubes in the cytoplasmmdashfragment duringcell division but otherwise form an interpenetrating structure

Actin and microtubule polymers are another example of active cellular components Both are stiff filaments that con-stantly assemble and disassemble with lifetimes on the order

of seconds or minutes Individually they are transitory but collectively they self-organize into much longer-lived sub -cellular structures Microtubules organize themselves into thespindle that segregates chromosomes during cell division and actin organizes itself into a specialized layer of proteinsknown as the cortex which has different functions such ascontrolling cell shape in different cells Figure 2b shows anelectron-tomography reconstruction of a mitotic spindle of a nematode embryo Caenorhabditis elegans Just 10 μm long the spindle is a dense network of 10000 microtubules thatholds the chromosomes at its center and reaches throughoutthe cytoplasm

From one viewpoint the cell is the ultimate multiscale system with scores of interlocking components From another its subcellular structures such as the nucleus spindle and en-doplasmic reticulum are active materials that can be concep-tualized at a coarse-grained level Indeed the perspective forbest viewing the cell and its contents depends on the problemat hand

An interest of ours is how after fertilization DNA frommother and father come together and become positioned in themiddle of the cell Figure 2c shows snapshots of that processAt its start (t = 0 s) the DNA is located inside membrane-boundorganelles called pronuclei with arrays of stiff microtubulesattached to the male pronucleus Those structures come to-gether to form the pronuclear complex (t = 45 s) which mi-grates through the cytoplasm to the cellrsquos center where it ro-tates to align with the cellrsquos long axis (t = 105 s) There thepronuclear membrane breaks down and the mitotic spindle

a b c

15 microm 35 microm

FIGURE 1 IMAGING SUBCELLULAR FLOWS (a) An 1857 illustration of the flow of granules in the cytoplasm of a crawling amoeboidcell Pelomyxa palustris as observed by microbiologist Franz Schulze (Adapted from ref 12) (b) The instantaneous flow velocities (yellowarrows) of the cytoplasm in a Drosophila oocyte The motion is driven by microtubules and molecular motors (Adapted from ref 13) (c) A three-dimensional fluid-dynamics simulation of flows produced during the positioning of the pronuclear complex in a nematode embryo (Caenorhabditis elegans) In this model positioning is achieved using molecular motors that walk along microtubules5 (Courtesy of Ehssan Nazockdast)

FLUID DYNAMICS


forms (t = 285 s roughly corresponding to figure 2b) To under-stand pronuclear migration itrsquos useful to think of the cyto-plasm as a viscoelastic continuum

Accounting for forcesLiving cells are small and their internal motions are slow Thattruism simplifies the fluid dynamics Consider a body of lineardimension l moving with speed u in a cell whose cytoplasmhas a viscosity μ and density ρ The cytoplasmrsquos Reynolds num-ber the ratio of inertial to viscous forces is given by

(1)

Take the transport of an l asymp 10 μm pronucleus in C elegans as aparadigmatic example The cellrsquos cytoplasm has a viscosityabout 1000 times that of water and the embryorsquos pronucleusmoves through it with speed u asymp 01 μms That yields aReynolds number of approximately 10minus9 a situation in whichinertial forces are so low theyrsquore irrelevant

Such a low Reynolds number puts us in the regime fa-mously discussed by Edward Purcell2 in his paper ldquoLife at lowReynolds numberrdquo in which the velocities of objects follow frombalancing applied forces with viscous drag Whereas Purcellrsquosarticle explored small things living and moving in fluids herewe explore fluids moving in small things

One can think of the cellrsquos interior crowded with various or-ganelles and polymer networks as an active multiphase ma-terial that is both viscous and elastic3 For viscoelastic materialsa common measure of the relative importance of viscous toelastic responses is the Weissenberg number

(2)

where γ is a typical strain rate of the flow and τe is the relaxationtime of the elastic medium The migration of the C eleganspronuclei to the center of the cell is slow and happens over thetime scale of minutes So we estimate γ ~ ul ~ 10minus2 sminus1 The faintelastic response4 of the embryonic cytoplasm produces a relax-ation time τe ~ 1 s Thus Wi = 10minus2 ≪ 1 and for the purposes ofstudying positioning the cytoplasm can reasonably be treatedas a purely viscous medium

That approximation considerably simplifies the response ofthe cytoplasm to motion but itrsquos not universal Some eventssuch as ruptures of the nuclear membrane can be more rapidin cancer cells they happen on time scales of seconds Such fastmotions yield much higher values of Wi In general a cellrsquos in-ternal mechanics are cell-type and cell-stage dependent and itsmechanical responses are a function of both the length andtime scales of the measurement1 and of where in the cell the responses are measured4

Viscous forces alone lead to unusual features of cell me-chanics One is the nature of drag on moving objects At lowReynolds number drag scales with length as μlu Consider a microtubule having the same linear dimension as the pro -nuclear payload l = 10 μm and moving at the same speed Thatmicrotubule is just 25 nm in diameter which makes its surfacearea 11000 that of the pronucleus yet itrsquos drag is nonetheless14 of the pronucleus drag Thatrsquos because drag in that regimeresults from objects shearing the viscous fluid even thin objectscarry along and distort large regions of fluid (compare the leftand right sides of figure 3a) Our daily experience by contrast

Re = =inertial force ~ ρu l2 2

viscous force ~ μul

ρul

μ

Wi = γτe

middot

10 microm

b

ChromosomesAstral MT Spindle MT Kinetochore MT

0 s

45 s

105 s

285 s

ca

500 microm

FIGURE 2 COMPLEX COMPONENTSSTRUCTURES DYNAMICS This electron-tomography reconstruction(a) reveals the tightly packed composition of a 3-microm-wide regionof a human pancreas cell False colors indicate distinct classes of itsorganelles and cytoskeletal filaments(Adapted from ref 14) (b) An electron-tomography reconstruction of a nematode (Caenorhabditis elegans)mitotic spindle The structure is approximately 20 microm long Approximately 10000 microtubules(MT) are shown many connected to chromosomes in the center theinset clarifies the spindlersquos structure(Adapted from ref 15) (c) Fluorescencemicroscopy of microtubules in a single-cell C elegans embryo revealsa sequence in which two clumps ofparent DNA (dark circles) migrate meet rotate together as a fusedpronuclear complex and form thespindle (Adapted from ref 16)


FLUID DYNAMICS

takes place mostly in high-Reynolds-number environmentswhere just by moving we push fluid aside with the surface areawe present to it

That microtubules carry considerable fluid is underappre-ciated Scientists unfamiliar with fluid mechanics tend to thinkof the cytoplasm as simply a source of local friction on objectsmoving through it That assumption can lead to the wrong con-clusion that drag is an additive force It would imply for in-stance that the drag on a 10000-microtubule arraymdashroughlythe number in the C elegans pronuclear complexmdashwould be10000 times the drag on a single microtubule It turns outthough that comoving microtubules in an array collectivelyentrain fluid or equivalently move in each othersrsquo wake Thusdrag on the assembly is on the order of the drag on just a single microtubule

When microtubules or other objects carry fluid around con-finement by cell boundaries can have a surprisingly strong effect To appreciate the consequences imagine the single-cellC elegans embryo as a sphere of diameter L asymp 50 μm filled withpurely viscous cytoplasm and the pronucleus as a sphere ofdiameter l asymp 10 μm A classical solved problem in a low-Reynolds-number flow is the drag on one sphere moving in alarger sphere It depends fundamentally on only the geometricratiomdashthe so-called confinement number Cn the ratio of thepayload size to the confining dimension

(3)

For Cn = 02 it takes almost twice the force to move the payloadat a given speed as to move it in an unbounded medium Therequired force increases quickly with Cn and is the cost of re -directing the induced cytoplasmic flow at the cell periphery Fur-ther accounting for the presence of attached microtubule arrayswhich effectively increases l and hence Cn results in a 100-foldincrease in drag5

Internal forces drive internal flowsMotions and hence flows in cells are often driven by motorproteins that move in the cytoplasm Many of those motor pro-teins can sequentially bind to move along and unbind frompolymers such as microtubules and actin That generates forces

between the motor and polymer and the resulting directedmotion is only possible because motor proteins transducechemical energy supplied by adenosine triphosphate (ATP)which is held out of equilibrium with its breakdown productsby the cellrsquos metabolism Energy available from the disequi-librium of ATP and its breakdown products acts as a batterythat motor proteins can tap to perform mechanical workWhen abutting or attached to other objects the polymers them-selves can also generate active forces through their assemblyor disassembly

For cellular forces intuition can be highly misleading Forexample the magnitude of a conservative force such as gravityor electrostatics on an object can be measured by determiningthe experimentally applied opposing force that halts the ob-jectrsquos motion But that procedure fails utterly in the case ofmotor proteins To understand why keep in mind that the forcebetween a polymer and an isolated motor is balanced by dragthrough the surrounding fluid

Because of the motorrsquos small size (about 1 nm) and speed(about 1 μmmin) that drag is about 1 femtonewton Applyingan opposing force to the motor slows it down but a motorprotein typically halts only when it is subject to forces in thepiconewton range thousands of times greater than the forcesit exerts in the absence of additional applied forces The upshotis that forces exerted by motor proteins in situ cannot be mea -sured simply by subjecting the proteins to external forces

Inside cells motor proteins exert forces in different waysand those forces have consequences for the flows they produceFor example the mechanism of pronuclear migration is currentlydebated Some researchers argue that the pronuclear complexis positioned by motors anchored at the cortex others arguethat itrsquos positioned by motors in the cytoplasm The first modelpostulates that motors fixed on the cell boundary reel in themicrotubules they walk on Microtubules would thus be trans-ported to the cell surface and drag along the pronuclear com-plex and the neighboring fluid

In the second model by contrast motors in the cytoplasmbind and walk along microtubules and thereby induce fluid toflow along the microtubulesrsquo surfaces By momentum balancepushing fluid backward propels the microtubule forward sothat it ldquoswimsrdquo through the cytoplasm Measurements of fluid

Cn =l

L

FIGURE 3 (a) FLOW LINES (white) are produced by a sphere (right) and a microtubule (left) both moving upwards at the same speed Thecolor field represents the magnitude of the fluid velocity which subtly differs for the sphere and microtubule but is greatest (white) aroundboth surfaces (b) Fluid flow lines are created by a force dipole (center) (c) A detail from a simulation of the chaotic flows created by asuspension of mobile force dipoles the color field represents flow vorticity (Image courtesy of David Stein)


flow should provide a clear way to determine which model iscorrect Roughly speaking motors at the cortex would drivefluid in the same direction as the pronuclei whereas motors in the cytoplasm would drive fluid in the opposite directionBut the detailed predicted flow patterns are complicated by theincompressibility of the fluid and confinement from the cellboundary5

The movement of motor proteins on polymers in the cyto-plasm is ubiquitous in cell biology The resultant forces betweenthe motors and polymers must be equal and opposite Thusthe forces those objects exert form force dipoles (see figure 3b)Collections of objects exerting dipolar forces on fluids can pro-duce spectacular phenomena such as self-organized swirlingflows called active turbulence shown in figure 3c and confine-ment-induced coherent flows Both types have been observedin cell extracts and reconstituted mixtures of motor proteinsand microtubules6

That purified biological components can robustly create suchintricate patterns of flow is remarkable In vitro systems maychart a promising pathway to investigate principles relevantfor subcellular biological fluid mechanics and self-organizationAnd they open the possibility of engineering artificial activematerials with lifelike properties The dynamics of such collec-tions of objects exerting dipolar forces on fluids can be charac-terized in terms of the activity number

(4)

Here c and σ are the concentration and magnitude of forcedipoles respectively u is the characteristic fluid velocity pro-duced by a dipole and L is the system size When A exceeds acritical value the system changes to a state of active turbulence7

When polymers are anchored to the cell boundary as for theactin cortex motors can move along the boundary The forceacting on the motor from the boundary is balanced by the dragforce generated by the motor moving through and shearingthe fluid that surrounds it When such motors are at low den-sity they move relatively independently of each other At highdensity the parcels of fluid dragged by the motors overlapwhich can lead to large-scale flows in the cytoplasm

At what density of motors does the large-scale flow occurObjects in a fluid create flow perturbations on the scale of theirown size so for the fluid to be entrained collectively the spac-ing between motors must be comparable to their size Motorproteins are a few nanometers in size but typically micronsapart in cells With that spacing motors are unlikely to producecoherent flowsmdashthat is to carry the fluid collectively But inmany contexts motor proteins transport not just themselvesbut also micron-sized organelles The additional fluid movedby the large payloads can be sufficient to produce bulk flowsin cytoplasm Known as cytoplasmic streaming such bulk flowsare observed in a wide variety of systems including plant cellsand egg cells8

Diffusion helps and is hinderedThe flows generated by cytoplasmic streaming or active turbu-lence can in principle advect and mix objects in the cytoplasmIn addition to being carried by the flows mobile objects canalso collide randomly with neighboring molecules and give

rise to diffusive motion The size of cells and the mobile objectsin them greatly influences the efficacy of transport processes1The Peclet number

(5)

quantifies the relative importance of the rates of advection anddiffusion for an object moving at a velocity u over a length lwhere D is the diffusion coefficient The dependence of Pe onthe length scale over which transport occurs shows that advec-tion becomes increasingly important as length scales get largerThe size dependence may explain why cytoplasmic streamingis most commonly seen in large plant and algal cells but itdoesnrsquot solve the problem of how cellular reactants becomewell mixed That process might require complex flows thatstretch fluid to bring reactants into close proximity or to en-hance the effective diffusion through dispersion (reference 8see also the article by Michael Brenner and Howard StonePHYSICS TODAY May 2000 page 30) The flows produced by ac-tive turbulence are well suited for such mixing and may be use-ful in microfluidic devices

Models and methodsThe numerical modeling of flows in cells is still in its infancyThe cellsrsquo internal complexity makes the task computationallydemanding and reliant on fluid models to organize and accountfor the complexity The most basic fluid model is the incom-pressible homogeneous Stokes equations of a low-Reynolds-number flow which is solved as a boundary-value problemusing the forces and velocities of objects moving in the fluidIn other settings the Stokes equations are replaced by simplerporous-medium approximations or forcing terms are added tomodel the effects of immersed structures

Usually the modeling approach depends on the particularcell in question and on the goal of studying it In the case ofpronuclear positioning microtubules are the objects throughwhich forces are transmitted to the pronuclear complex Sim-ulating microtubules introduces a few challenges For startersmicrotubules are extremely thin and yet microns long Theyare also numerous typically with 10thinsp000 in an array whichcomplicates the flow geometry Whatrsquos more motor proteinscan readily bend the microtubules which further complicatesthe geometry And finally microtubules are transitory typicallyliving just 30 s and they exhibit various lengths because of theirconstant assembly and disassembly That heterogeneity leadsto a wide range of time scales because relaxation times frombending are length dependent

We have recently developed specialized methods to simu-late the fluid dynamics of microtubule systems The methodsuse Green functions of the three-dimensional homogeneousStokes equations to derive integral equations on immersed andbounding surfaces Because of the thinness of microtubules the integrals over their surfaces can be further reduced to 1Dline integrals These reductions in dimensionality together withother techniques simplify the simulation of the flows gener-ated by thousands of moving bending microtubules over timescales (typically tens of minutes) that are relevant to pronucleartransport5

Those methods are optimal for capturing the detailed hy-drodynamics of microtubules but they remain limited in scope

A = =active forces ~ σc σcL

uμviscous force ~ μuL

Pe =ul

D


and can handle at best a few thousand filaments In other casesresearchers are interested in hundreds of thousands or evenmillions of cytoskeletal filaments Capturing the hydrodynam-ics of such problems requires approaches that ldquocoarse-grainrdquothe fine details In the example of Drosophila oocytes molecularmotors moving on wall-anchored microtubules set up com-plex flows believed to move signaling molecules about the celland help establish its internal asymmetry or polarity9 Biophysi-cists have studied the flows produced by the motorsrsquo activity bynumerically solving a Stokes equation augmented by a coarse-grained forcing term that represents the directed motion of themotors moving along microtubules (see figure 4a)

For other problems it is useful to treat the internal mechan-ics of the cell as arising from a coarse-grained possibly multi-phase complex material In that general approach the cytoplasmis modeled as particle-packed or visco(poro)elastic and drivenby internal forces from molecular motors or polymerization4The biophysics of cell motility is one area in which considerableefforts have been made to simulate such coarse-grained mod-els10 A related area is that of cell-shape changes which arisenaturally in cell motility and in cellndashcell interactions

Finally not all cellular fluid dynamics takes place in the cy-toplasmic volume DNA can move in seemingly coordinatedways in the cellrsquos nucleus Recent modeling and simulationssuggest that those motions may arise from force dipoles createdby molecular machines acting within the fluidic nucleoplasm11

as shown in figure 4b

A to-do listFor understanding subcellular fluid mechanics this article hasoutlined five dimensionless numbers the Reynolds Weissenbergconfinement activity and Peclet It is easy to think of otherssuch as the volume fractions of mobile particles and of poly-mer meshes and arrays As the study of subcellular fluid me-chanics matures the dimensionless numbers postulated to de-scribe it will grow no doubt to rival the variety commonly usedto categorize the dynamics of Earthrsquos oceans and atmosphere

Much work remains Biophysicists still lack validated theo-ries of the collective properties of active fluidic materials and

a framework that explains how large-scale behaviors such asactive turbulence depend on the behaviors of constituent mol-ecules The contribution of fluid mechanics to distinct sub -cellular phenomena is also yet to be worked out And its absenceis a major hurdle to understanding the mechanics and behav-iors of those phenomena

Progress will surely require new conceptual and technicaladvances The potential payoff for such work is huge Not onlywill it provide insight into beautiful and mysterious processesthat epitomize the unique properties of living matter it willalso have practical import Just as an understanding of plane-tary fluid flows (see the article by Erdal Yiğit and Alexander SMedvedev PHYSICS TODAY June 2019 page 40) has enabledpredictions of the weather and led to transformative economicand social improvements an understanding of cellular fluidflows will allow scientists to predict the behavior of biologicalprocesses And those predictions in turn are likely to have atransformative effect on the treatment of diseases

We appreciate Reza Farhadifarrsquos help in preparing the figures

REFERENCES1enspK Luby-Phelps Int Rev Cytol 192 189 (2000)2enspE M Purcell Amer J Phys 45 3 (1977)3enspA Mogilner A Manhart Annu Rev Fluid Mech 50 347 (2018)4enspC Garzon-Coral H A Fantana J Howard Science 352 1124

(2016)5enspE Nazockdast et al Mol Biol Cell 28 3261 (2017)6enspD Needleman Z Dogic Nat Rev Mater 2 17048 (2017)7enspM J Shelley Annu Rev Fluid Mech 48 487 (2016)8enspR E Goldstein J-W van de Meent Interface Focus 5 20150030

(2015)9enspP K Trong et al Elife 4 e06088 (2015)

10enspI S Aranson ed Physical Models of Cell Motility Springer (2016)11enspD Saintillan M J Shelley A Zidovska Proc Natl Acad Sci USA

115 11442 (2018)12enspL Pickens Organization of Cells and Other Organisms Clarendon

Press (1960)13enspS Ganguly et al Proc Natl Acad Sci USA 109 15109 (2012)14enspB J Marsh et al Proc Natl Acad Sci USA 98 2399 (2001)15enspN Lindow et al Methods Cell Biol 145 45 (2018)16enspT Shinar et al Proc Natl Acad Sci USA 108 10508 (2011) PT

FLUID DYNAMICS

a b

FIGURE 4 SIMULATING COMPLEX FLOWS (a) A computational model of flows driven by motors moving on microtubules in a Drosophilaoocyte (Courtesy of Joumlrn Dunkel image adapted from ref 9) (b) Simulations of chromatin (the functional form of DNA) in the nucleus beingmoved by internal molecular machines At left the chromatin self-organizes into an aligned structure At right is the nucleoplasmic velocityfield (Adapted from ref 11)

The stormy fluid dynamics of the living cell

Flow lines from a simulation of pronuclearmigration (Courtesy of Tamar Shinar)





Daniel Needleman

and Michael Shelley







C








a b c

15 microm 35 microm


FLUID DYNAMICS




(1)




(2)






ρul

μ

Wi = γτe

middot

10 microm

b


0 s

45 s

105 s

285 s

ca

500 microm



FLUID DYNAMICS




(3)








Cn =l

L






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b






Daniel Needleman

and Michael Shelley







C








a b c

15 microm 35 microm


FLUID DYNAMICS




(1)




(2)






ρul

μ

Wi = γτe

middot

10 microm

b


0 s

45 s

105 s

285 s

ca

500 microm



FLUID DYNAMICS




(3)








Cn =l

L






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b









a b c

15 microm 35 microm


FLUID DYNAMICS




(1)




(2)






ρul

μ

Wi = γτe

middot

10 microm

b


0 s

45 s

105 s

285 s

ca

500 microm



FLUID DYNAMICS




(3)








Cn =l

L






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b





(1)




(2)






ρul

μ

Wi = γτe

middot

10 microm

b


0 s

45 s

105 s

285 s

ca

500 microm



FLUID DYNAMICS




(3)








Cn =l

L






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b



FLUID DYNAMICS




(3)








Cn =l

L






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b






(4)






(5)








Pe =ul

D

















FLUID DYNAMICS

a b


















FLUID DYNAMICS

a b


Documents

The stormy fluid dynamcis of the living cellThe stormy fluid dynamicsof the living cell Flow lines from a simulation of pronuclear migration. (Courtesy of Tamar Shinar.)