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Oxford Reference. Publications Pages Publications Pages. Recently viewed 0 Save Search. Your current browser may not support copying via this button. Submitted 29 May Accepted 30 Jun First published 04 Jul Download Citation. Author version available. Download author version PDF. Request permissions. Dynamic density functional theory of protein adsorption on polymer-coated nanoparticles S.
Social activity. Search articles by author Stefano Angioletti-Uberti. Typically, such a particle moves ballistically at short times, but eventually changes its orientation, and displays random-walk behaviour in the long-time limit. Here we verify this prediction quantitatively by constructing bacteria that swim with an intensity-dependent speed when illuminated and implementing spatially-resolved differential dynamic microscopy sDDM for quantitative analysis over millimeter length scales.
Einstein predicted and Perrin verified that, in a gravitational field, the equilibrium number density of a dilute dispersion of colloidal particles varied with height z according to. The verification of Eq.
Nevertheless, statistical mechanics stipulates that the spatially dependent dynamical coefficient D 0 z cannot appear in the equilibrium density distribution, and Eq. Active colloids 2 , particles that dissipate energy to propel themselves, form an important class of active matter 3 , 4.
Such dissipative states of matter, which include all living organisms, are intrinsically non-equilibrium, and give rise to new physics. Consider a system of run-and-tumble particles RTPs. Suppose the run speed of such particles is spatially dependent, v r. A purely mechanical derivation is also possible 7. Qualitatively, Eq. In a spatially varying illumination pattern, cells accumulate in the darker regions, generating contrast. Quantitatively, however, Eq.
Moreover, its theoretical derivations do not include hydrodynamic interactions, so that its applicability to real systems is open to doubt. Here, we investigate Eq. Differential dynamic microscopy DDM 12 has previously been shown to reliably measure swimming speed and density of motile bacteria on uniform samples 10 , Each E.
When all flagella rotate counterclockwise seen from behind , they bundle and propel the cell. Our E. These are living analogues of synthetic light-activated active colloids 20 , Given that higher cell densities occur in darker regions with lower swimming speed, Eq. We now proceed to test it quantitatively. Demonstration of spatial-resolved differential dynamic microscopy.
Projecting light patterns onto a sample of light-powered E. Indeed, a recent attempt to verify Eq. Details of other strains we used are given in the methods section. Strictly speaking, a non-interacting limit does not exist for bacterial suspensions We observed collective motion in our E. It was not possible to work below this limit because of an increasing fraction of cells trapped in circular trajectories due to hydrodynamic interactions with walls of the sample chamber 24 that did not explore the whole sample compartment, hindering relaxation towards a steady state.
A stepped speed pattern was developed Fig. Error bars are s. If Eq. If the density of non-motile cells is constant throughout the experiment see Supplementary Note 2 for justification , i. Equations 6 — 9 together predict the density of motile cells in the two half-planes:. One possible explanation is the emergence of collective motion with associated local nematic ordering 25 , which would invalidate the derivation of Eq.
Nevertheless, Fig. The data are noisier, but show the same trends. The same quantities as in Fig. Qualitatively the behaviour is the same as for the higher density, but the data are noisier due to the overall weaker signal. See Supplementary Figs. For both low density filled symbols as well as higher-density datasets open symbols Eq. Error bars show s. We performed experiments using the stepped light pattern at other cell densities and also using an additional smooth swimming strain DM1.
Experiments on such time scales are impractical due to mechanical and biological stability issues. Thus, we only have direct evidence for the validity of Eq.
This suggests that the use of a series of thin stripes would give more clear-cut results unencumbered by kinetic issues. We found that this was indeed the case. This is possible not only because of the length scale reduction, but also because swimmers can enter say a high-intensity region from low-intensity regions on both the left and right.
Error bars represent s. Error bars show propagated s. Interestingly, experiments using low light intensities which gave low swimming speeds proved less successful, because at very low intensities we found a noticeably higher percentage of non-motile cells in the sample than at high light intensities Supplementary Fig. This led to a spatial variation in the non-motile density Supplementary Fig.
We end by explaining why we did not use motility wild-type run and tumble strains for our experiments. Their motion randomises much more rapidly than smooth swimming mutants, which would have significantly alleviated the non-steady-state issue for the stepped intensity pattern.
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