Kinetic Analysis Reveals the Identity of Aβ-Metal Complex Responsible for the Initial Aggregation of Aβ in the Synapse

The mechanism of Aβ aggregation in the absence of metal ions is well established, yet the role that Zn2+ and Cu2+, the two most studied metal ions, released during neurotransmission, paly in promoting Aβ aggregation in the vicinity of neuronal synapses remains elusive. Here we report the kinetics of Zn2+ binding to Aβ and Zn2+/Cu2+ binding to Aβ-Cu to form ternary complexes under near physiological conditions (nM Aβ, μM metal ions). We find that these reactions are several orders of magnitude slower than Cu2+ binding to Aβ. Coupled reaction-diffusion simulations of the interactions of synaptically released metal ions with Aβ show that up to a third of Aβ is Cu2+-bound under repetitive metal ion release, while any other Aβ-metal complexes (including Aβ-Zn) are insignificant. We therefore conclude that Zn2+ is unlikely to play an important role in the very early stages (i.e., dimer formation) of Aβ aggregation, contrary to a widely held view in the subject. We propose that targeting the specific interactions between Cu2+ and Aβ may be a viable option in drug development efforts for early stages of AD.

The problem with the Aβ competing against a zinc indicator for Zn 2+ is that the concentration of Aβ and zinc indicator must be in a 20 fold excess of the concentration of Zn 2+ for the experiment to be under pseudo-first-order conditions. Given that the literature K d is in the range of 1 µM to 100 µM, even stoichiometric concentrations of Aβ would cause rapid Aβ aggregation, more so with a 20 fold excess. Therefore we measured Zn 2+ competing with Cu 2+ for labelled Aβ.
We take a simplified reaction model to describe the kinetic competition experiments as shown below The reaction of Aβ·Cu → Aβ + Cu 2+ is ignored as the rate is approximately 0.5 s −1 , 1 much slower than the observed binding rates (20 s −1 to 100 s −1 ). The reaction Aβ·Cu + Zn 2+ → Zn·Aβ·Cu is also ignored as the rate constant is 3×10 3 M −1 s −1 (determined in main text; Kinetics of Zn binding to Aβ-Cu) , which gives a rate of 0.6 s −1 at the highest Zn 2+ concentration used in the experiments. The reaction Aβ·Zn + Cu 2+ → Zn·Aβ·Cu is ignored because in order for the reaction to have a rate of at least 1 s −1 , a rate constant of 2×10 6 M −1 s −1 or greater is required. This is deemed unlikely given that the rate constants for the reaction Aβ·Cu + Zn 2+ → Zn·Aβ·Cu and Aβ·Cu + Cu 2+ → Cu·Aβ·Cu are 3×10 3 M −1 s −1 and 1×10 5 M −1 s −1 respectively. The later reaction is therefore also too slow to participate (0.05 s −1 ). By the same reasoning further reaction of Zn 2+ with Aβ·Zn is also ignored. The fast phase of the curves obtained from the stopped flow measurements were fitted to where K d is the equilibrium dissociation constant (K d = k -zn /k zn ) of Aβ-Zn complex.

Simulation of the reactions between metal ions and Aβ in the synaptic cleft by coupled reaction-diffusion equations
The simulation was based on a simplified model of the synaptic cleft with a height of 20 nm. It is technically a 3D simulation, but we assume that the cleft width is infinite so that the diffusion of metal ions released is not restricted to the typical synaptic width of a few hundred of nanometers. Simulation in confined space is not appropriate as metal ions would not be able to escape. We also assume that the space is both translationally and rotationally symmetric, so only 1D is needed in polar coordinates. Metal ions (30 µM Cu 2+ or 300 µM Zn 2+ ) were assumed to release into the centre of the synapse via 40 nm diameter vesicles and react with 3 nM Aβ in the cleft. The diffusion coefficients used are D Zn 2+ =D Cu 2+ =650 nm 2 µs -1 , 2 D Aβ =304 nm 2 µs -1 and D HSA =61 nm 2 µs -1 . 3 The synapse height adopted is in line with the literature. [4][5] To simulate the diffusion of metal ions and Aβ in the synapse, Fick's second law with constant diffusion coefficient was used, where φ is a scalar field of the concentration, t is the time and D is the diffusion coefficient.
In cylindrical coordinates, this equation can be simplified to To solve this equation numerically, it needs to be discretized. which forms a set of equations for the next time step. Due to a pole at ‫ݎ‬ ൌ 0, the function needs to be moved off the axis by mapping r to ‫ݎ‬ ߜ‫ݎ‬ 2 ⁄ in the implementation, thus Eqn 8 becoming The boundary condition at ‫ݎ‬ ൌ 0 is At the edge of the disc ሺ‫ݎ‬ ൌ ܴሻ, there are two options. The first option is to set the concentration of the ring outside the disc to 0, that is This will cause the molecules to "leak" off the edge in the simulation. Although this is suitable for metal ions diffusing from the centre, it is not suitable for a homogeneous concentration of Aβ as Aβ will diffuse off the disc. To counteract this, the edge of the disc was connected to itself by mapping ߮ ோାఋ ௧ାఋ௧ to߮ ோିఋ ௧ାఋ௧ , making the boundary conditions To incorporate reactions into the model, the rate of the change of the concentration due to reaction ܴሺ߮ሻ is added to the diffusion Eqn 5, thus becoming For a simple reaction, if the scalar fields of concentrations of A, B & C are φ, ψ and χ respectively, the changes in concentration due to the reaction will be ܴሺ߮ሻ ൌ ݇ ୭ ߯ െ ݇ ୭୬ ߮ο߰, ܴሺ߰ሻ ൌ ݇ ୭ ߯ െ ݇ ୭୬ ߮ο߰ and ܴሺ߯ሻ ൌ ݇ ୭୬ ߮ο߰െ݇ ୭ ߯ (15) Hence discretising these and applying the Euler method, the equations to be solved are Boundary conditions can be produced similarly as above.
In the reaction-diffusion equations (Eqn. 14-18) the reaction and diffusion parts may be treated separately in each time step simplifying the implementation, as the reaction only depends on the concentrations at a particular r and the diffusion depends on all r but only for the concentrations in one field. It can therefore be easily parallelized using OpenMP. 6 The discretized solution for φ is of the matrix form where M is a matrix of coefficients related to diffusion (similarly for ψ and χ). This gives the order of the procedure as the following: get the concentration fields, perform the reactions, and then calculate the concentrations at the next time step from diffusion. The concentrations of the next time step can be obtained from M is tridiagonal, therefore solving for ߮ ௧ାଵ may be done more efficiently than via matrix inversion. The gsl_linalg_solv_tridiag from the GSL 7 was then used.
For the simulation, the time steps were chosen to increase exponentially after the second time step t 1 (t 0 =0), defined by where T is the maximum time of the simulation, t i is the first non-zero time step, N t is the number of time steps, and i is an integer from 0 to N t . Logarithmic time steps allow the long timescale behaviour to be seen without exponentially increasing processing power. For the space steps, linear spacing is used as the central finite difference method requires ‫ݎߜ‬ to be constant across the simulation.
To simulate the case of periodic pulsed release of metal ions during neurotransmission, the concentration of metal ions at the centre (20 nm radius) of each release was reset to initial concentration. The time step was set to where T p is the pulse period, N s is the number of steps per pulse, and mod is the modulo operator. Then when ݅modܰ s ൌ 0, the central 20 nm of the simulation is reset to the initial concentration of metal ions.
The simulation code was written in C++, using the GSL and the OpenMP API, and complied with GCC. 8