Spatiotemporal dynamics of self-assembled structures in enzymatically induced agonistic and antagonistic conditions

Predicting and designing systems with dynamic self-assembly properties in a spatiotemporal fashion is an important research area across disciplines ranging from understanding the fundamental non-equilibrium features of life to the fabrication of next-generation materials with life-like properties. Herein, we demonstrate a spatiotemporal dynamics pattern in the self-assembly behavior of a surfactant from an unorganized assembly, induced by adenosine triphosphate (ATP) and enzymes responsible for the degradation or conversion of ATP. We report the different behavior of two enzymes, alkaline phosphatase (ALP) and hexokinase (HK), towards adenosine triphosphate (ATP)-driven surfactant assembly, which also results in contrasting spatiotemporal dynamic assembly behavior. Here, ALP acts antagonistically, resulting in transient self-assemblies, whereas HK shows agonistic action with the ability to sustain the assemblies. This dynamic assembly behavior was then used to program the time-dependent emergence of a self-assembled structure in a two-dimensional space by maintaining concentration gradients of the enzymes and surfactant at different locations, demonstrating a new route for obtaining ‘spatial’ organizational adaptability in a self-organized system of interacting components for the incorporation of programmed functionality.

To get compositional change of each component with respect to time the above-mentioned equations were solved using Python 3.0. To get the compositional change of individual component, the rate constants used in equation S1-S4 were fixed and so formed ordinary differential equations were solved using odeint function of Numpy packages [S1] . The code used for solving above-formed equations is shown in note S1. For instance, the parameters used are shown in table S1 and composition of dimeric assembly (S2T and S2P) and total assembly formed (S2T + S2P) is shown in figure S1. In addition to this, to get a broader view of effect of rate constants on individual components, parameters were varied (table S1) and the relative distribution of composition is shown in fig. S1.

E. DFT study
All quantum chemical calculations pertaining to surfactant and surfactant with ATP, ADP, G6P were performed by Density Functional Theory (DFT) method, using Gaussian 09 software. [S3] The geometries of stationary points were optimized by minimizing energies with respect to all geometrical parameters without imposing any molecular symmetry constraints using 6-311G basis set and for dispersion corrections, WB97XD, functional was used. Frequency calculations showed the absence of any imaginary frequency modes which confirmed that the optimized structure is an energy minimum. The solvent correction was done using conductor polarized continuum model (CPCM).    In Fig. S17

J. Spatiotemporal study of assembly formation using fluid dynamics
To study the spatiotemporal evolution of S2T and S2P (mentioned in section B) under flow condition, a computational model was designed using MATLAB R2019b, FEATool Multiphysics. Here, we collectively studied the transport mechanism of both reactive and non-reactive species involved in system along with their rate of formation and deformation using computational fluid dynamics [S4-S6] . For this purpose, we designed our system so that T was distributed evenly on the squared grid while either S or S, and E were added from opposite boundaries as shown in figure S30 and figure 4a and 4d in main manuscript.
For describing the one-dimensional distribution of reactive species over space with time we have used Fick's second law of fluid dynamics so that ( , ) = 2 2 − ± S10

S28
where C = the concentration of the species at position x and time t, = diffusion coefficient, = the linear flow velocity, and = rate of change of species, while the '±' symbol denotes the rate of formation and deformation of species.
Following above equation, we rewrote the equations for each component of our system incorporating previously mentioned mass balance equations (S5-S9) in place of . Apart from this, Because of horizontal placement of grid, we considered the effect of diffusion on species movement and neglected the velocity term in equation S10. Now, to solve our system of coupled differential equation under flow conditions, we first designed a square grid space containing 14,816 grid points and 294,912 triangular grid cells, an example is shown in figure S29. For initial conditions, we assumed that T (1 μM) is evenly distributed on grid. It is to be noted that here all the dimensions are unitless and we have assumed the unit of concentration and time as µM and hour, respectively for ease of understanding. Also, we assumed that diffusion rate of all assembled states is equal for simplification of equations. Firstly, we studied when monomer S was added from one side (boundary 4) of grid, while all other boundaries were left empty ( fig. S30a). For solving the coupled equations, we marked the boundary conditions for all the species in accordance with Dirichlet and Neumann boundary conditions at all boundaries [S7] . For this case, the only equation to be followed was equation S1, as E was absent. Sample code containing used equations is shown in note S2.
For quantifying, S2T assembly formation by holding Neumann condition at boundary 4 and Dirichlet condition at boundary 2. The parameter fixed for S2T at boundary 4 was 1 μM/h, while 0.2 μM/h at boundary 2. Also, S followed Dirichlet condition as 1 μM/h, and 0.5 μM/h at boundary 4, and boundary 2, respectively. While solving this system of equation, over 5 h with 0.02-time steps following Crank-Nicolson time-step scheme, we observed that as time increases S2T formation increases. The maximum S29 S2T concentration can be seen at boundary 4 which gradually decreases as move towards boundary 2 ( fig. 4b+c in main text, supplementary video SV1). Note S2: Apart from this, when we added S from boundary 4 and E from boundary 2, the set of equations holding were (S1) - (S4). The set of equations used in Matlab are shown in note S3. Here also, the parameters for S, R, and S2T were same and this set of equations was also solved in similar way as mentioned in previous paragraph. Additionally, the rate of product formation and product-driven dimeric assembly (S2P) formation was added at respective boundaries.
For S2P formation over time, the Neumann boundary parameter was fixed at 0.5 μM/hour at boundary 2. While solving this set of equations over similar time constraint, we observed formation of S2T formation at boundary 4 which decreases gradually as we move towards boundary 4, while S2P assembly formation was more at boundary 2 which gradually decreases as we move towards boundary 4 (fig. 4d+e, supplementary video SV1, SV2).            Table S13. Calculated P-values (in parenthesis) to determine the significance level of the counting of fluorescent particles between different zones for the image Fig. 5c of the main manuscript for the data points taken at 5 min. Here ATP was evenly distributed in all zones and from zone A, surfactant and from zone E, HK enzyme was added. Here AB signifies comparison between zone A and B, similarly BC between zone B and C and so on.   (a and c) A comparative study between theory and experiment when only S was added from the left side when T is distributed through the space (for theoretical modelling) and surfactant was added from the left side when ATP was distributed under the cover slip (for experiment). (b and d) Both assembled structure S2T (from theory) and number of self-assembled unit for surfactant and ATP assembly (from experiment) were plotted across zones A to E at the end point of simulation or experiment.

AB
From this data, decreasing trend of structures from zone A to E has been observed in both cases. Fig. S40. (a and c) A comparative study between theory and experiment when S was added from the left side and E from the right when T is distributed through the space (for theoretical modelling) and surfactant was added from the left side and ALP was added from right when ATP was distributed under the cover slip (for experiment). (b and d) Both assembled structure S2T (from theory) and number of self-assembled unit for surfactant and ATP assembly (from experiment) were plotted across zones A to E at the end point of simulation or experiment.

S41
From this data, decreasing trend of structures and also much lower number structure in all zones compared to without E have been observed (see Fig. S39). The population decreased from zone A to E has been observed in both cases, where in experiment zone A and B showed higher number of structures, compared to zone C to E. Similar kind of trend was also observed in theoretical trend. Fig. S41. (a and c) A comparative study between theory and experiment when S was added from the left side and E from the right when T is distributed through the space (for theoretical modelling) and surfactant was added from the left side and HK was added from right when ATP was distributed under the cover slip (for experiment). (b and d) Both assembled structure S2T (from theory) and number of self-assembled unit for surfactant and ATP assembly (from experiment) were plotted across zones A to E at the end point of simulation or experiment.

S42
From this data, almost equal number of structures in all zones have been observed, both in experiment and theory.