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molecular machines are protein molecules which can transform chemical energy into ordered internal mechanical motions . the classical examples of such machines are molecular motors kinesin and myosin , where internal mechanical motions are used to transport cargo along microtubules and filaments . many enzymes operate as machines , using internal conformational motions to facilitate chemical reactions . other kinds of machines , operating as ion pumps or involved in genetic processes , are also known . moreover , artificial nonequilibrium nanodevices , similar to protein machines , are being developed @xcite . the cycle of a protein machine is typically powered by chemical energy supplied with an atp molecule . when an atp molecule binds to a protein machine , the initial protein - atp complex is out of equilibrium and the process of ordered conformational relaxation of this complex towards its equilibrium state starts . when this state is reached , atp is converted into adp and this product molecule leaves the complex . the emerging free protein molecule is out of equilibrium and another conformational relaxation process , returning the free protein to its original equilibrium state begins . thus , the cycle consists of two conformational motions . it is important to note that the forward and back conformational motions inside each cycle do not coincide - they correspond to relaxation processes in two different physical objects , i.e. the free protein and the protein - atp complex @xcite . recently , much attention has been attracted to microscale swimmers operating at low reynolds numbers . generally , it can be shown that any physical object , cyclically changing its shape in such a way that the internal forward and back motions are different , propels itself through the liquid @xcite . elementary models of such swimmers , constructed by joining together mobile links @xcite or by connecting a few spheres by several actively deformable links @xcite , have been considered . typically , the concept of molecular swimmers is applied to explain active motion of bacteria and other microorganisms . however , individual macromolecules that operate actively as machines are also capable of self - motion @xcite    if a swimmer is attached to some support , preventing its translational motion , it exhibits force acting on the mechanical support . the generated force is equal to the stall force which needs to be applied to a swimmer to prevent its translational motion . when many swimmers are attached to a distributed support , their collective operation produces mechanical pressure acting on the support . this situation is , for example , encountered for molecular machines representing active protein inclusions ( such as ion pumps ) in biological membranes . it has been shown that the combined effects of the mechanical forces generated by the machines and their lateral motions inside the membrane can lead to membrane instabilities @xcite . the aim of this paper is to investigate a different situation involving molecular machine populations . we consider hypothetical machines which are floating on top of a thin liquid layer and thus represent _ active surfactants _ ( fig.[fig1 ] ) . because the considered surfactant molecules are microswimmers , they generate local pressure applied to the liquid - air interface and proportional to their local concentration . additionally , they are subject to surface diffusion and , as surfactants , also modify local surface tension . our main analytical result , supported by numerical simulations , is that , if the surface density of machines is sufficiently high and if the rate of supply of chemical energy to the machine population exceeds a threshold , the equilibrium flat interface becomes unstable and hydrodynamical flows inside the liquid layer spontaneously develop . this instability is accompanied by spatial redistribution of floating machines and their collective motions over the surface . it leads to the emergence of a special kind of surface turbulence . such phenomena can provide the basis for active microfluidics , where hydrodynamical motions in liquid layers are induced and controlled by molecular machines located at the surface . the next two sections are devoted to the derivation and the stability analysis of a model of such molecular swimmers attached to the surface of a thin liquid film . a collection of numerical results is presented in section iv . finally , a model and numerical results are discussed , and some estimates of the forces of such molecules are given . the possible experimental realization is also discussed . population of identical molecular machines floating on the surface of a thin liquid film will be considered . these machines perform cycles of conformational changes ( with a characteristic time @xmath0 ) enabled by the supply of energy through atp molecules present in the liquid . we are interested in macroscopic collective effects and do not specify the details of machine operation . it will be assumed that , on the average , each machine generates the force which is applied to the air - liquid interface . this average force acts along the normal direction ( the forces in the lateral direction vanish after averaging because of the rotational diffusion ) . each machine cycle is initiated by binding of an atp molecule and the average force is proportional to the cycle frequency . assuming that binding of atp molecules follows the michaelis - menten law , the average force per machine molecule is    @xmath1}{k_{atp}+[atp]}. \label{atp}%\ ] ]    here , @xmath2 is the maximal force under the atp saturation conditions , @xmath3   $ ] denotes the atp concentration in the liquid , and @xmath4 is the characteristic concentration at which saturation begins .    because of the cycles , the floating machines produce additional pressure acting on the air - liquid interface . this pressure @xmath5 is proportional to the local surface concentration of the machines and the average force generated by an individual machine , i.e. @xmath6 , where @xmath7 is the force per mol of molecules ( @xmath8 where @xmath9 is the avogadro number ) . note that the machines are still sufficiently well separated one from another and possible synchronization effects of their cycles ( cf . @xcite ) are therefore neglected .    before proceeding to the detailed formulation of the model , we want to outline the origin of the expected surface instability . suppose that the average force , generated by machines is directed upwards and therefore the machines are pulling the liquid up in the vertical direction ( fig . if machine concentration is increased in some region , the pulling pressure is higher in this region , inducing local rise in the liquid film thickness . this however leads to lateral hydrodynamical flows which are directed inwards and bring even more machines into the region . as a result , the positive feedback , responsible for the instability , is established . note that surface diffusion of floating machines and capillary forces are acting in the opposite direction , suppressing the instability . thus , it is observed only if the average force @xmath10 generated by a machine is large enough , setting a threshold in terms of the energy supply rate ( i.e. , of the atp concentration in the solution ) . the instability is not possible when the force is downwards directed , thus inducing local depressions of the liquid layer ( fig . 2b ) , then , in contrast to the previous case , hydrodynamical flows remove machines from the depression , restoring the equilibrium flat film . temporal evolution of the surface concentration field @xmath11 is described by the equation :    @xmath12    where @xmath13 is the lateral fluid velocity and @xmath14 is the surface diffusion . floating proteins represent surfactants and reduce the surface tension of the interfaces ( see @xcite ) . to take this effect into account , we assume that the surface tension coefficient @xmath15 decreases linearly with the machine concentration , @xmath16    hydrodynamical flows , induced by the gradients in machine concentration , should be further considered . we assume that the liquid layer is so thin that the lubrication approximation , typically employed in microfluidics @xcite , is justified . as shown in appendix a , the evolution equation for the local hight @xmath17 of the interface has the form    @xmath18    where the local pressure is @xmath19 and @xmath20 is the viscosity of the fluid . determining the lateral flow velocity at the interface ( see appendix a ) and substituting it into the evolution equation for the surface concentration , a closed set of two partial differential equations is obtained . explicitly , the considered dynamics of thin films with active surfactants is described by equations :    @xmath21 \nonumber\\ &   + \frac{1}{\mu}\sigma_{c}\vec{\nabla}\left (   \frac{h^{2}}{2}\vec{\nabla } c\right )   , \label{dth}%\end{aligned}\ ] ]    @xmath22 \nonumber\\ &   + \frac{1}{\mu}\sigma_{c}\vec{\nabla}\left (   ch\vec{\nabla}c\right ) + d\nabla^{2}c . \label{dtc}%\end{aligned}\ ] ]    note that we have assumed that the active surfactant is insoluble @xcite .    for subsequent analysis , it is convenient to write these equations in the dimensionless form . the ( equilibrium ) liquid layer thickness @xmath23 will be used as the length unit , time will be measured in units of @xmath24 , and local concentration @xmath11 in units of the equilibrium machine concentration @xmath25 . changing the variables as @xmath26 , @xmath27 , @xmath28 and @xmath29 , we obtain @xmath30   \right ) \nonumber\\ &   + \frac{a}{2}\vec{\nabla}\left (   h^{2}(\vec{\nabla}c)\right )   , \label{pth2}%\end{aligned}\ ] ] @xmath31 \nonumber\\ &   + a\vec{\nabla}\left [   ch\vec{\nabla}c\right ]   + d\nabla^{2}c . \label{ptc2}%\end{aligned}\ ] ]    the new model equations include only three dimensionless parameters    @xmath32 the parameter @xmath33 specifies the characteristic strength of floating machines as the surfactant species ( decreasing the local surface tension of the interface ) . the parameter @xmath34 specifies the magnitude of the pressure generated by the cycling molecular machines ; it is controlled by the rate of energy supply to the system . finally , @xmath35 is the dimensionless diffusion coefficient of floating machines .    comparing the last terms in equation ( [ ptc2 ] ) , we notice that spreading of floating machines , induced by changes in the surface tension , has the same functional form as surface diffusion . if relative variations of the film thickness and machine concentration are small ( @xmath36 ) , the effective diffusion coefficient of this process is @xmath33 . our estimates below in sect . v indicate that the genuine diffusion constant @xmath35 of floating machines is typically much smaller than the effective diffusion constant @xmath33 . having this in mind , we retain the terms including the coefficient @xmath35 in our analytical investigations , but put @xmath37 when numerical simulations of the model are performed . results with non - zero diffusion coefficient have been also obtained but they are very close to the results with @xmath37 . as shown in appendix a , the dimensionless horizontal ( @xmath38 ) and vertical ( @xmath39 ) velocities of the liquid film at hight @xmath40 and horizontal spatial location @xmath41 can be found as @xmath42 @xmath43 when the fields @xmath44 and @xmath45 are known . both velocities vanish at the bottom of the liquid film , at @xmath46 , because of the no - penetration and no - slip boundary conditions imposed there . the model always has the stationary uniform state @xmath47 . to perform the linear stability analysis of this state , small perturbations @xmath48 @xmath49 and @xmath50 @xmath49 in the form of plane waves with the wavenumber @xmath51 are introduced . linearizing evolution equations with respect to small perturbations @xmath52 and @xmath53 and solving the linearized equations , growth rates @xmath54 of such perturbations are obtained ,    @xmath55   ^{2 }     \nonumber\\   &   + 2 \left (   1-a\right )   \left (   \frac{a}{2}-\frac{b } { 3}\right )   k^{2 } ) ^{1/2 }   .\nonumber\end{aligned}\ ] ]    it can be easily checked that , in absence of the energy supply ( @xmath56 ) , both rates are real and negative , so that the equilibrium flat film is stable as should be expected . when the parameter @xmath34 is increased , the instability develops at @xmath57 where @xmath58 above the instability threshold , traveling plane waves with the wavenumbers @xmath51 near @xmath59 , @xmath60 and the frequencies near @xmath61 @xmath62 are growing.the fastest growing mode with @xmath63 and @xmath64 is characterized by the growth rate@xmath65 = \frac{3}{32 }   \frac{~\left (   b - b_{c}\right)^{2}}{1-a}. \label{rate}%\ ] ]    figure 3 shows dependences @xmath66   $ ] and @xmath67   $ ] at three values of the parameter @xmath34 below and above the instability boundary . while the fastest growing mode at @xmath68is always oscillatory , with @xmath69   \neq0 $ ] , above the instability boundary the system also always has some standing growing modes with the wavenumbers @xmath51 smaller than @xmath70 . as the boundary is approached from above , both the wavenumber @xmath70 and the frequency @xmath71 decrease and vanish at @xmath57 . the region with two real modes @xmath72 and @xmath73 always lies below @xmath74 ; it shrinks and vanishes at @xmath75 . similar long - wavelength instabilities have previously been discussed for other conservative systems ( see @xcite ) . to investigate the behavior of the system in the nonlinear regime above the instability onset , numerical simulations have been performed . equations([pth2],[ptc2 ] ) were integrated using the semi - implicit method ( see appendix b ) for a one - dimensional system using periodic boundary conditions . as the initial condition , the flat interface with a uniform machine distribution was chosen and small random initial perturbations were applied . our main observation is that the instability development results in the emergence of a complex spatiotemporal regime which can be described as _ interface turbulence_. this regime is characterized by spontaneous appearance of _ traveling machine clusters _ ( i.e. , of spatial regions where the local machine concentration is increased ) and of the accompanying local interface modulations . figure([fig4 ] ) gives an illustration of the turbulent regimes observed at different different distances from the instability threshold . the local film thickness @xmath45 is displayed here in gray scale depending on the spatial coordinate ( the vertical axis ) and time . in the slightly supercritical regime at @xmath76 , the transient development of standing waves is first observed , which is then followed by the emergence of an irregular pattern of traveling and colliding waves . at larger deviations from the critical point , the transients are faster and the irregular wave dynamics appears soon after the instability onset . the characteristic spatial scale of the turbulence decreases with the control parameter @xmath34 , consistent with the predictions of the linear stability analysis . the characteristic velocity of traveling waves is also growing with @xmath34 . figure [ fig5 ] shows snapshots of computed turbulent patterns at different deviations from the instability threshold . both the interface profiles and the corresponding concentration distributions are presented here . again , a decrease of the characteristic wavelength of the irregular spatiotemporal patterns under an increase of the control parameter @xmath34 can be noticed . moreover , we see that the characteristic amplitude of the waves grows with @xmath34 . to illustrate the directions of wave propagation , arrows are placed in the insets of this figure , where examples of colliding and traveling waves are shown . these arrows indicate the directions in which the respective concentration and interface profile maxima are shifting at the next time moment . note that the motions of machine clusters ( i.e , of the local concentration maxima ) are typically guiding the motions of surface bumps . temporal transients leading to turbulent patterns are characterized in fig . [ fig6 ] . to construct it , we have determined the maximum and the minimum values @xmath77 and @xmath78of the film thickness as function of time , starting from the initial moment . their difference @xmath79 ( where @xmath80 is the dimensionless time ) can be chosen to describe the amplitude of the developing patterns . as seen in the two top panels in fig . [ fig6 ] , this amplitude first grows ( exponentially ) and the undergoes saturation . the transient is shorter for the larger deviation from the threshold ( @xmath81 ) and the final mean amplitude of the turbulent pattern is also then larger .    to estimate the characteristic wavenumber @xmath82 of the developing spatial patterns , their fourier transforms were computed and the positions of dominant maxima in the spatial power spectra were determined at different time moments . as seen in the two bottom panels in fig . [ fig6 ] , these wavenumbers do not significantly change during the transients . the characteristic wavelengths of the developed turbulent patterns are therefore not much different from those of the critical modes . figure [ fig7 ] shows temporal dependences of the hight @xmath83 and the concentration @xmath84 at a fixed point of the system in the final turbulent state . the characteristic time of the oscillations is clearly different for the two values of the control parameter . both properties fluctuate around some mean values . the fluctuations are more rapid farther away from the instability threshold .    finally , fig . [ fig8 ] displays the dependence of several selected statistical properties of the final turbulent state on the deviation @xmath85 from the critical point @xmath86 . the characteristic wavenumbers @xmath87 and amplitudes @xmath52 have been computed as described above . time averages of these properties and statistical dispersion of the data are shown in figs . [ fig8 ] . the characteristic transient times @xmath88 ( fig . [ fig8]c ) have been estimated by fitting the initial computed time dependence for @xmath89 to the exponential law , @xmath90 . the characteristic frequency @xmath91 of the patterns ( fig . [ fig8]d ) is estimated by computing their temporal power spectra and determining the positions of the dominant maxima . solid curves in fig . [ fig8 ] give fits of the simulation data to the power laws predicted by the linear stability analysis . according to equations ( [ wavenumber ] ) and ( [ frequency ] ) , @xmath92 @xmath93 and @xmath94 . the transient time is determined by the rate of growth as @xmath95   $ ] and , according to equation ( [ rate ] ) , we expect that @xmath96 . for the mean amplitude @xmath52 of the patterns , the quadratic fit @xmath97 has been applied . we see that the statistical properties of nonlinear patterns in the developed turbulence regime are still in good agreement with the respective predictions based on the linear stability analysis . can predicted interface instabilities be experimentally observed ? this question has both chemical and physical aspects . on the chemical side , it is known that proteins may indeed represent surfactants and thus float at the air - water interface ( see . e.g. , @xcite ) . although we can not give here a specific example , it seems plausible that some protein machines also belong to this class . moreover , other protein machines , including molecular motors , can probably be made floating by chemical modification , i.e. by attaching to them a hydrophobic group . the physical question is whether the propulsion forces generated by individual protein machines would be sufficient to induce the considered interface instability . according to equation ( [ boundary ] ) , the instability is reached when @xmath98 with @xmath99 . taking into account the definitions of dimensionless properties @xmath33 , @xmath34 and @xmath35 , the instability condition implies that the propulsion force @xmath100 generated by a single machine , must exceed the threshold    @xmath101    where @xmath23 is the film thickness , @xmath25 is the surface concentration of floating machines , @xmath14 is their surface diffusion constant , @xmath20 is the liquid viscosity , @xmath9 is the avogadro number , and @xmath102 specifies the surfactant capacity of the considered biomolecules ( the surface tension coefficient @xmath15 depends as @xmath103 on their concentration @xmath11 ) .    for numerical order - of - magnitude estimates , we choose @xmath104 g @xmath105s@xmath106 ( the viscosity of water ) , @xmath107 @xmath108s@xmath106 ( characteristic diffusion constant of large biomolecules in water solutions @xcite ) . based on the experimental data given in ref . @xcite , we take furthermore @xmath109 . both terms in equation ( [ force ] ) are inversely proportional to the liquid film thickness @xmath23 and the smaller critical forces are therefore expected for the thicker layers . note , however , that this result is based on the lubrication approximation for thin liquid films , requiring that the film thickness is much smaller than the characteristic wavelength of the flow patterns . as we have seen , the characteristic wavelength of the interface turbulence in the considered system depends on the distance from the critical point ( cf . equation ( [ wavenumber ] ) ) and , in principle , can be arbitrarily large sufficiently close to the instability threshold . taking into account experimental limitations making it very difficult to work too close to the threshold , we choose however @xmath110 as the characteristic maximum film thickness .    concentration @xmath25 of floating proteins enters only into the second term in equation ( [ force ] ) . this term , depending on surface diffusion of floating machines , can be neglected in comparison with the first term , if the condition @xmath111 with @xmath112 holds . substituting numerical values , we get @xmath113 . thus , already for very low surface concentrations of proteins , effects of their diffusion can be neglected in the considered problem .    neglecting diffusion effects , the critical propulsion force of a single machine is estimated as @xmath114 note that it does not depend on the protein concentration . substitution of the numerical values into equation ( [ force1 ] ) yields @xmath115 . can a single molecular machine generate the hydrodynamical propulsion force of that magnitude ? direct measurements of molecular propulsion forces are still not available . it is known that molecular motors , such as myosin or kinesin , can generate mechanical forces of about @xmath116 @xcite , but these data refers to the molecules moving along microtubules and filaments , not to the swimmers .    for a simple example , we consider the elementary three - body purcell swimmer @xcite . its characteristic propulsion velocity is about @xmath117 , where @xmath118 is the displacement per single cycle and @xmath119 is the characteristic time of the cycle ( cf . the viscous friction force of an object of linear size @xmath120 that moves with velocity @xmath121 through the liquid is by the order - of - magnitude @xmath122 . thus , propulsion at velocity @xmath123 would require the propulsion force about @xmath124 . choosing @xmath125 nm , @xmath126 and @xmath127 ms and considering water as the liquid , we obtain for the molecular propulsion force the rough estimate of @xmath128pn . similar estimates are obtained for other known elementary swimmers . note that the actual average propulsion force @xmath10 of a machine additionally depends on the frequency of machine cycles and the angle with respect the interface , therefore , according to equation ( [ atp ] ) , this gives the estimate of the maximum propulsion force under the energy - supply saturation conditions and the most efficient orientation of the machine .    while molecular propulsion forces are quite small , according the above estimates they would still be sufficient to induce the film instability and lead to the interface turbulence . therefore , we conclude that the experimental observation of the predicted effects is _ principally possible_.    the experiments aimed at detecting instabilities of thin liquid layers induced by floating actively operating machines would allow to directly estimate actual propulsion forces generated by particular biomolecules . investigations of such nonequilibrium hydrodynamical systems would be very important from the perspective of _ active microfluidics _ , where active motions in thin liquid layers are produced by floating biomolecular propellers .    financial support from the eu marie curie rtn `` unifying principles in nonequilibrium pattern formation '' and from the german science foundation collaborative research center sfb 555 `` complex nonlinear processes '' is acknowledged . we are grateful to u. steiner , d. barbero and u. thiele for valuable discussions . we consider the situation when the film thickness @xmath23 is the smallest characteristic length of the system . in this case , the lubrication approximation can be used which corresponds to an expansion in the small parameter @xmath129 , with @xmath130 being the characteristic wavelength of the patterns in the lateral direction . hydrodynamical equations in the lubrication approximation are simplified and take the form ( see , e.g. , @xcite ) @xmath131 with the incompressibility condition @xmath132 here , @xmath133 is the vertical component of the fluid velocity and @xmath13 is its horizontal component ; @xmath134 is the differential operator acting only on the horizontal coordinates . note that , as follows from these equations , pressure @xmath135 is constant along the vertical direction . the boundary conditions at the bottom of the film , in contact with the solid support , are @xmath136 at the free air - liquid interface @xmath137 , the balance of horizontal and vertical forces should separately hold , implying that @xmath138 where @xmath15 is the surface tension coefficient and @xmath5 is the additional pressure produced by floating active molecules . conservation of the total film volume implies moreover that the equation . @xmath139 should hold .    integrating equations.([stokes ] ) and taking into account the boundary condition ( [ p ] ) , the horizontal velocity of the flow can be determined , @xmath140   .\ ] ] the vertical velocity is then obtained by using the incompressibility condition , @xmath141    substituting these expressions into equation ( [ interface ] ) and integrating over @xmath142 , the final form of the interface equation is derived , @xmath143 while simple explicit euler methods are frequently employed in numerical investigations of nonlinear reaction - diffusion models , such methods easily become numerically unstable when hydrodynamical microfluidics equations are considered . therefore , a specially constructed numerical integration method has been employed in our study . below , its brief description is provided . there are two nonlinear differential equations to solve , one for the thickness @xmath45 and the other for the concentration @xmath44 of the surfactant . we define vector @xmath144 and formally write both equations as @xmath145 the matrix operator @xmath146 can be decomposed as @xmath147 into its linear ( @xmath148 ) and nonlinear ( @xmath149 ) parts . they are defined as@xmath150    to compute @xmath151 at the next time step @xmath152 , the mixed semi - implicit method is employed : @xmath153 thus , the implicit method is used to determine the contribution from the linear part in time @xmath152 and the explicit method is employed to compute the nonlinear part contribution . the calculation of the inverse differential matrix operator @xmath158 is most conveniently performed by transforming the equation to the fourier space . here , it is important that the term @xmath159 is determined before applying the fourier transformation . our simulations using this numerical method have been performed only in the one - dimensional case , @xmath160 and @xmath161 . in two dimensions , the method becomes complicated and such simulations have not been undertaken .
thin liquid films with floating active protein machines are considered . 
 cyclic mechanical motions within the machines , representing microscopic swimmers , lead to molecular propulsion forces applied to the air - liquid interface . 
 we show that , when the rate of energy supply to the machines exceeds a threshold , the flat interface becomes linearly unstable . as the result of this instability , 
 the regime of interface turbulence , characterized by irregular traveling waves and propagating machine clusters , is established . 
 numerical investigations of this nonlinear regime are performed . 
 conditions for the experimental observation of the instability are discussed .