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Molecular dynamics made visible through mathematics

A clever algorithm allows scientists to capture the ultra-fast movements of molecules and other dynamic processes occurring in the nanocosmos to within a quadrillionth of a second. In developing this method, a team of international researchers has made a crucial advance in analysing dynamic processes. Their work, which is now being published in the journal “Nature”, opens up a comparatively simple way of determining the sequence of events during elementary reactions with a very high temporal resolution.

The team led by Abbas Ourmazd, a professor at the University of Wisconsin in Milwaukee, has developed a special mathematical technique able to extract accurate dynamical information from data recorded at highly uncertain time points. This has been demonstrated on ultrashort timescales with data obtained from so-called free-electron lasers (FEL). The results are confirmed by quantum-mechanical simulations performed by Leading Scientist Robin Santra (Universität Hamburg, DESY, CFEL) and his colleagues.

Thanks to the new algorithm scientists can see the time stamp sharper than ever before. Credit: Allie Kilmer/University of Wisconsin – Milwaukee

“This method has unbelievable potential,” explains Santra, who is also a member of CUI. It provides completely new insights into the events taking place during numerous ultra-fast chemical and biochemical reactions, including electrochemical applications and industrial processes – fields in which scientists have until now been confined to speculating about the temporal order of events occurring on the microscopic level. “Dynamic time measurements using FELs are subject to extreme uncertainty,” Santra explains. “This new method of data analysis can increase their accuracy by a factor of 300 – which is astonishing.”

Chemical reactions and the movements of biomolecules occur incredibly quickly and are therefore not accessible to unaided observation. They take place within femtoseconds, quadrillionths of a second, during which light travels about 2% of the width of human hair. Until now, no effective method has been available for observing such molecular processes in detail. Although modern X-ray free-electron lasers do permit femtosecond exposures, these cannot be converted directly into moving images of dynamic processes, but simply produce a series of snapshots taken at different times during the process under investigation.

The exact time at which these images are recorded cannot be determined, however, because when scientists want to study a particular reaction, they first trigger it using a pulse of optical laser light and then take a snapshot shortly afterwards, using an X-ray laser pulse. This process destroys the sample and the same reaction has to be triggered in a new, nearly identical sample. This time, the X-ray pulse is triggered slightly later during the reaction – and so on. As a result, the scientists obtain countless snapshots, which they then have to string together, like in a flip book. But the exact sequence of the X-ray laser snapshots is corrupted by timing uncertainty, a fluctuation that experts refer to as jitter.

Twelve Million Dimensions

“This temporal uncertainty is a curse in many areas of science,” says Ourmazd. “You have lots of data, but no accurate time stamps.” Yet, in order for the snapshots to document the sequence of events during a reaction with femtosecond accuracy, the optical and X-ray lasers must be very precisely synchronised. “None of the experimental solutions known to us so far has been able to achieve a temporal resolution better than around 14 femtoseconds, and most measurements are limited to 60 femtoseconds or longer,” says Santra.

Ourmazd and his team therefore opted for a different approach: they developed a mathematical algorithm that would allow them to extract information from the available data with a temporal accuracy of one femtosecond. For this purpose, the individual snapshots with the fuzzy time stamps are represented as individual points in a highly multidimensional space – in the paper now published, this space has around twelve million dimensions. A mathematical pattern recognition procedure is then used to reduce the number of dimensions by looking for curved surfaces on which the points lie. Tracking long sequences of snapshots leads to a one-dimensional curve on which the points are correctly ordered in time. Because if the individual points only differ in terms of a single parameter, in this case time, they must necessarily form a curved line in the space under consideration.

In the study now published, the scientists used their algorithm on data that had been collected by a research group led by Philip Bucksbaum at the SLAC National Accelerator Laboratory in California. In 2010, Bucksbaum and his team studied the dynamics of doubly-charged ions of nitrogen (N2), using the X-ray free-electron laser LCLS (Linac Coherent Light Source) at SLAC. The scientists created these unusual nitrogen ions by firing X-rays at the molecules. The resulting ions are also formed in the Earth’s atmosphere, due to the constant bombardment by high-energy cosmic radiation. The result of the experiment was a large number of snapshots of different vibrational modes that occur in intact and ruptured nitrogen molecules, the temporal order of which could not be clearly ascertained. With the help of their algorithm, Ourmazd and his colleagues have now managed to determine the vibrations of the molecules to an accuracy of one femtosecond. This allowed them to reconstruct the dynamic behaviour of the nitrogen molecules with a precision increased by a factor of 300.

Revolutionary analytical technique

Santra and his team at CFEL then carried out a quantum-mechanical calculation of the processes involved, and confirmed the accuracy of one femtosecond. “We concluded this from the fact that the extracted periods of the oscillation correspond to our quantum-mechanical calculations with exactly this level of precision,” says Santra. And not just that: it was only through the simulation calculations carried out by Santra’s team that the scientists were even able to explain what was causing the vibrations observed in the experiment, what they mean, and why and when doubly-charged nitrogen molecules break apart.

The new technique for analysing data will not only allow future experiments to be analysed more precisely. Existing data can also be re-examined. The only requirement is that sufficient data must be available. This is not easy when it comes to examining three-dimensional structures, as the researchers point out. In crystallography, for example, where even a single snapshot requires a tremendous number of X-ray exposures, in order to obtain a statistically significant set of data. “But perhaps this problem will also be solved by the European XFEL,” says Santra. The 3.4 kilometre long X-ray free-electron laser, which is currently under construction in the west of Hamburg, will produce 100 times more pulses per unit of time than existing FELs.

“This method has what it takes to revolutionise the research conducted at FEL facilities,” says Santra. And it offers a huge advantage too: instead of resorting to complex technical solutions, it makes clever use of mathematical operations. “This approach is not only simpler, but also more successful, because the results are much more exact,” says Santra. The physicist envisages numerous possible applications. “This method can be used as a far more precise tool in all areas in which we would like to know what matter is doing – viewed dynamically on short timescales.” This includes enzyme reactions in biology and chemistry, but also studying unusual states of matter, such as those occurring inside planets and stars. Abbas Ourmazd, the project leader, goes even further and is hoping to use his algorithm to more accurately calculate the timing of an even wider range of processes, such as past climatic events.

R. Fung, A.M. Hanna, O. Vendrell, S. Ramakrishna, T. Seideman, R. Santra and A. Ourmazd
„Dynamics from noisy data with extreme timing uncertainty“
Nature, 2016
DOI: 10.1038/nature17627