Molecular swimmers
Mechanical propulsion of molecules in solution
Swimming at Low Reynolds number
The start of my PhD focused on the possibility of ‘molecular swimmers’: the ability of molecules to move through solution at rates faster than diffusion. Whether or not this is possible (or at least, feasible) remains an open question: molecules are very small, and molecular-scale fluid dynamics at low Reynolds number are completely different to what most of us are more familiar with at larger length scales. The classic treatment of these differences is E. M. Purcell’s very readable Life at Low Reynolds Number, which introduces the memorable “scallop theorem” describing the impossibility of swimming using reciprocal motion (like a scallop!) at low Reynolds number. Instead, non-reciprocal (=time-asymmetric) swimming motions are needed such as corkscrew flagella or semi-flexible cilia. Some non-reciprocal swimmers with rigid bodies are shown below.
Non-reciprocal molecular swimmers
What might a non-reciprocal molecular swimmer look like? It would necessarily be a molecular motor: breaking time-reversal symmetry means directional cycling through (at least three) states, which is really another way of saying “uses energy to do mechanical work”. I had the idea of making a simple molecular-mechanical swimmer build around two photoswitchable linkers driven indirectly by FRET from a dye appended to one end, with the distant-dependent FRET efficiency used to bias cycling towards first isomerising the near linker and only then isomerising the far linker:
Cool idea, right? Unfortunately, I quickly realised it didn’t make much sense. Let’s assume that each isomerisation drives the molecule forward by distance \(d\), and \(k\) isomerisation events occur per unit time \(t\). The average displacement per unit time \(\langle x \rangle\) will then be given by
\[\langle x_{swim} \rangle = dkt\]How does this compare to diffusion? Diffusive transport follows
\[\langle x_{diff}^2 \rangle = 6Dt\]and ‘swimming’ will thus be comparable to or greater than diffusion for
\[dk > \sqrt{6D}\]For a small-ish molecule in organic solution, let’s say \(D = 10^{-10}~\mathrm{m^2 s^{-1}}\). It seems reasonable that the ‘step size’ for an isomerising molecule must be smaller than its length, so let’s charitably assume that our molecules are perfectly efficient swimmers wiht a length of 1 nm, giving us \(d = 10^{-9}~\mathrm{m}\). The minimum isomerisation rate to drive measurable swimming motion must then be:
\[k > \frac{\sqrt{6\times10^{-10}}}{10^{-9}}~\mathrm{s^{-1}}\\\] \[k > \sqrt{6}\times 10^4~\mathrm{s^{-1}}\]That’s tens of kHz of photoisomerisation per molecule! From memory, the NMR experiments I was running with approximately 100 mW in-situ irradiation were delivering something closer to 1 photon per molecule per second, so reaching 10+ kHz of isomerisation per molecule would have needed something like a megawatt of LED power which wasn’t going to happen any time soon. This intensity of light irradiation might have been possible in what would at that point be a lossy optical cavity, but I’ve got no idea how you’d be measure the diffusion of the molecules against such a bright background and in the presence of so much heating: sample heating was already a confounding artefact at the much lower powers I’d been operating at.
I don’t think that this sort of mechanical swimming makes much sense for molecules. Since \(D \propto 1/R^2\) and therefore \(\langle x_{diff} \rangle \propto 1/R\), while the velocity of mechanical swimming at a given frequency follows \(\langle x_{swim} \rangle \propto R\), the overall efficiency of swimming over diffusion is proportional to \(R^2\) and molecules are unfortunately just too small to make much headway.
Fortunately, by the time I figured this out I’d found a new and much more interesting approach to making molecules swim faster: intriguing reports of the “enhanced diffusion” of catalysts during chemical reactions.
Footnote: do molecular swimmers really require non-reciprocal motion?
As above, Purcell’s scallop theorem says that reciprocal motion can’t be used to swim at low Reynolds number because each back-stroke exactly undoes the displacement gained by each forward-stroke. This seems to be reasonable for micrometre-scale bacteria, but I don’t think it’s true for nanometre-scale molecules. The reason for this is that rotational diffusion (tumbling) of small molecules in solution completely scrambles their orientation over picosecond-nanosecond timescales, meaning that it’s extremely unlikely for a reciprocal back-stroke to exactly undo the translational displacement of the forward stroke. This means that reciprocal swimming with stroke frequency \(\omega_{stroke}\) should increase (diffusive) transport up to the point where tumbling at frequency \(\omega_{tumble}\) can no longer re-orient the swimmer between strokes. Larger low-Reynolds swimmers like bacteria presumably operate entirely within this regime where \(\omega_{tumble}\) is low, swimming is ballistic (directional), and swimming strokes must be non-reciprocal. Smaller swimmers (molecules?) operate in a different regime where \(\omega_{tumble}\) is high, swimming is diffusive (directionless), and it doesn’t really matter whether swimming strokes are directional or not.
