Gillespie/ODE

Introduction

Whereas phospholipids mainly aggregate into two-dimensional aggregates, i.e., bilayers, other molecules aggregate into one dimensional aggregates, i.e., filaments or supramolecular polymers. Examples of proteins forming fibrous aggregates are actin, microtubules, and amyloids. Here we study such filament formation using two synthetic molecules, namely BTA's and OPV's. Both of these molecules can form one dimensional helical supramolecular polymers, where the helicity can be both left-handed and right-handed.

Achiral monomers will form aggregates of both helicities in equal amount, while chiral information in the monomers results in a preference for one helicity in the aggregates. We study both the thermodynamic equilibrium of these supramolecular polymerizations, for instance showing the important role of the monomer pool in chiral amplification, as well as the kinetics, where we for instance observed that not always the thermodynamically most stable aggregate is formed first.

Pathway complexity

A first example is the supramolecular polymerization of the S enantiomer of OPV. Two of such molecules can form a very stable dimer, which can be the building blocks of a supramolecular polymerization. In a very low concentration solution, these building blocks will be present as monomers. When the concentration is increased, these monomers start aggregating. Aggregates smaller than some critical size (the nucleus), are assumed to be still rather unstructured and as such less favorable than larger, nicely helical structured, aggregates. As a result this aggregation process is very cooperative, yielding long supramolecular polymers.

The aggregation process can be modeled using the following scheme, where two types of aggregates (P and M) can be formed, representing the right-handed and left-handed helical aggregates respectively. In this model, new dimers can be formed from two monomers and existing supramolecular polymers can grow or shrink by successive monomer additions and removals, where the rate constants may differ between the two aggregate types and between before and after reaching the critical nucleus size (n).

The kinetics of this model can then be solved by integrating the set of differential equations describing these reactions. The excess of material in aggregates with one helical sense over the material in aggregates with the other helical sense can then for instance be compared with experimentally measured circular dichroism data of these systems. In this way the competition between different aggregation pathways could be unravelled [Nature 481, 492-496 (2012)].

Stochastic simulation

Instead of using differential equatons, the supramolecular polymerization process could also be modeled stochastically, for instance using kinetic Monte Carlo simulations. In the case of a supramolecular polymerization of a single monomer type X, the current state of the system is given by the number of monomers and the numbers of polymers and their lengths. Given these numbers of monomers and polymers and the reaction rates, the probabilities of all possible reactions can be calculated. When we descibe the polymerization again as a monomer addition/removal process, the possible reactions are:

At each iteration of the simulation, the probabilities of all possible reactions are calculated and one reaction is selected proportional to these probabilities and is executed. After a reaction has been selected, products are stoichiometrically incremented, reactants are decremented and the time by which the system advances is determined by randomly drawing a number from an exponential distribution with as parameter the sum of all reaction propensities. An example of a simulation of a system containing only 60 molecules is shown in a movie that can be started by pressing the left most picture below.

The other two pictures link to movies of isodesmic (middle) and cooperative (right) supramolecular polymerization. The difference between the two is that in the isodesmic case the rate constants for all monomer additions/removals are the same, whereas in the cooperative case these are still the same for aggregates larger than the critical nucleus size, but smaller for those smaller than this nucleus size. As can be seen from the movies, in the isodesmic case many short polymers are formed, whereas in the cooperative case fewer, but longer, polymers are formed. These movies thus provide a very nice illustration of the influence of cooperativity on supramolecular polymer growth.

Chiral amplification

It becomes even more interesting when we have two types of monomers that can copolymerize in single aggregates. For instance, when we have two enantiomers of a chiral molecule, where one enantiomer (R) prefers the right-handed helicity aggregates (P) whereas the other enantiomer (S) prefers the left-handed helicity (M). Possible reactions are then:

The movie below on the left hand site shows what happens if the monomers have a very high preference for one type of helical sense. In that case the two monomer types hardly mix and the two monomer types behave more or less as two separate systems. If the penalty for mixing in the opposite helicty aggregates is however relatively small compared to the energy gain of being in an aggregate mixed polymers are being formed. As can be observed from the movie on the right hand site below, in this case the ratio between material ending up in P and M helicity is much larger than the ratio of R and S present, something that is called the Majority Rules principle of chiral amplification.

Another example of chiral amplification is the so called Sergeants and Soldiers effect. Like the left-hand figure below shows, achiral monomers (black) aggregate in aggregates of both types of helicity in equal mounts. As can be seen in the movie at the right-hand site below, when a small percentage of chiral (red) monomers is added, finally only one type of helicity aggregates remains.

More information on these simulations and an alternative approach to describe such chiral amplification, which is based on a mass balance approach, is available in [Nature Communications 2, 509 (2011)]

Fischer-Tropsch

Another interesting example of reversible polymerization is the Fischer-Tropsch reaction, i.e., a heterogeneous catalytic reaction that converts synthesis gas (a mixture of carbon monoxide and hydrogen which can be derived from natural gas, coal, or biomass) into diesel quality liquid fuels. Taking the reversibility of the hydrocarbon chain growth into account using similar approaches as used for the supramolecular polymerizations, new insight into the behavior of this catalytic reaction is provided. This for instance not only shows that competition between reagent activation and product formation leads to a volcano-type dependence of the reaction rate on the reactivity of the catalyst surface, called the Sabatier principle. It also shows that the observed maximum in selectivity for longer hydrocarbon chains as a function of the reactivity of the catalyst surface can be explained by the reversibility of the chain growth [Angewandte Chemie 51, 9015 (2012)].

In follow-up studies we used the same microkinetics approach, based on quantum-chemical data of the relative stability of reaction intermediates and their corresponding elementary reaction rates, to gain further insight into the molecular mechanisms of the Fischer-Tropsch reaction. By solving the ordinary differential equations (ODEs) of the microkinetics models, without invoking any assumptions on particular rate controlling steps, we were able to discriminate between different possible molecular mechanistic models. This showed that not the CO insertion chain growth mechanism but the carbide chain growth mechanism is responsible for the chain growth in the Fischer-Tropsch reaction [ChemCatChem (2013)], and that kinetics operates within the so-called monomer formation kinetics limit, in which the activity is controlled by C-O bond activation that is the rate controlling step [J. Phys. Chem. C 117, 4488 (2013)]. Moreover, we were able to derive analytical formulae for the overall reaction rate as well as the selectivity for longer hydrocarbon chains of the different molecular mechanistic models. Direct integration of the ODEs furthermore allowed comparison with recent kinetic experiments on catalyst particle size dependence [Faraday Discuss. 162, 267-279 (2013)]. These insights are also important for the practical kinetics modeling of Fischer-Tropsch processes. A nice overview of all these new results and their implications for further study of the Fischer-Tropsch reaction are given in [Phys. Chem. Chem. Phys (2013)].

Contact

If you have any further questions on this research, please contact Peter Hilbers, Bart Markvoort or Tom de Greef.