Enzyme Kinetics

The following 6 reactions describe the antioxidant system inside the cortex and nucleus.

$$GP_{xr}+[H_2O_2]_{in}+H^+ \xrightarrow[]{k_1} GP_{xo}+H_2O ...(1)$$ $$GP_{xo}+GSH+H^+ \xrightarrow[]{k_2} [GS-GP_x]+H_2O ...(2)$$ $$[GS-GP_x] + GSH \xrightarrow[]{k_3} GP_{xr}+GSSG+H^+ ...(3)$$ $$NADPH \xrightarrow[GSR]{k4, K_{4M}} NADP^+ ...(4)$$ $$GSSG \xrightarrow[GSR']{k_5, K_{5M}} 2GSH ...(5)$$ $$[H_2O_2]_{out} \xrightarrow[]{k_5} [H_2O_2]_{in} ...(6)$$

Each reaction will be discussed in detail, and we will derive rate equations (Ault 1974,Pi 2004) .

Reaction 1:As hydrogen ions are numerous are negligible in the reaction, we will ignore it. By the law of mass action, the rate of this reaction is: $$r_1=k_1[GP_{xr}][H_2O_2]_{in}$$

Reaction 2: By the law of mass action, the rate of this reaction is: $$r_2=k_2[GP_{xo}][GSH]$$

Reaction 3: By the law of mass action, the rate of this reaction is: $$r_3=k_3[GS-GP_x][GSH]$$

Summary of Reaction 1-3:In these reactions, hydrogen peroxide is reduced to water. GSH is consumed to recycle the enzyme GPx back into reduced form, to neutralize more hydrogen peroxide.

Reaction 4: ByMichaelis-Menten kinetics and the Ping-Pong mechanism, the rate of this reaction, with rate constant k4, and Michaelis-Menten constant Km4, is: $$r_4=k_4\frac{[NADPH]}{K_{4M}+[NADPH]}$$

Reaction 5: ByMichaelis-Menten kinetics and the Ping-Pong mechanism, the rate of this reaction, with rate constant k5, and Michaelis-Menten constant Km5, is: $$r_5=k_5\frac{[GSSG]}{K_{4M}+[GSSG]}$$

Summary of Reaction 4-5: In these, GSSG is reduced back to form GSH, using the enzyme GSR. This is necessary for antioxidation to continue as reaction 1 is constantly using GSH, converting them to GSSG.

Reaction 6: By the Law of Passive Diffusion, the rate of diffusion into the cortex and lens is: is: $$r_6=k_6([H_2O_2]_{out}-[H_2O_2]_{in})$$

Differential Equations

We have six reaction rates derived from above. Now, we will form differential equations, where every time a species is used as a reactant, the reaction rate will be subtracted from the species’ derivative, while each time it is formed as a product, the reaction rate will be added to the species’ derivative. We will go through each species in detail:

Substituting the rate of each reaction, we get the following system of differential equations.

$$\frac{d[GP_{xr}]}{dt}=k_3[GS-GP_x][GSH]-k_1[GP_{xr}][H_2O_2]_{in}$$ $$\frac{d[H_2O_2]}{dt}=k_1([H_2O_2]_{out} - [H_2O_2]_{in})-k_1[GP_{xr}][H_2O_2]_{in}$$ $$\frac{d[GP_{x0}]}{dt}=k_1[GP_{xr}][H_2O_2]_{in}-k_2[GP_{xo}][GSH]$$ $$\frac{d[H_2O]}{dt}=k_1[GP_{xr}][H_2O_2]_{in}+k_2[GP_{xo}][GSH]$$ $$\frac{d[GSH]}{dt}=2k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}-k_2[GP_{xo}][GSH]-k_3[GS-GP_x][GSH]$$ $$\frac{d[GS-GP_x]}{dt}=k_2[GP_{xo}][GSH]-k_3[GS-GP_x][GSH]$$ $$\frac{d[GSSG]}{dt}=k_3[GS-GP_x][GSH]-k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}$$ $$\frac{d[NADPH]}{dt}=-k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}$$ $$\frac{d[GSR]}{dt}=k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}-k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}$$ $$\frac{d[GSR']}{dt}=k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}-k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}$$

Procedure

Listed outside collapsible

1. After building this differential equation model in Mathematica, we change the initial starting concentration of GSR, and numerically solve the equations.

2. The output will be a the concentration of H2O2 as a function of time. Integrating this function over 50 years using the definition in Model 1 returns the total crystallin damage.

3. We repeat this procedure from 0 to 100 uM of initial GSR, and graph the resulting crystallin damage against the initial GSR inputted (Figure 4).

Assumptions

In this experiment, we made the following assumptions:

1. Crystallin damage is linear with respect to both concentration and time, as evidenced by cysteine’s first order reaction with hydrogen peroxide (Domínguez-Vicent 2016). (see Model 1 for full explanation).
2. The amount of total GSR (in both forms) is constant. In reality, for the first days of the treatment the GSR level is being increased, so GSR levels will not be constant until 30 days of the treatment have passed (see Model 3). As we will discuss in Model 3, a one-time treatment will be made as soon as prevention is fully effective.
3. The cortex and nucleus of the eyes are indistinguishable and regarded as a single entity. As cataract damage can occur in both areas, we simplify the model by combining them to form a single system.
4. The amount of H2O2 in the aqueous humor (but the amount in the nucleus changes). As cellular respiration is constantly regenerating ROS and forming H2O2, there will constantly be H2O2 diffusing into the lens.
5. The amount of H2O2 in the aqueous humor is equal to the initial value of H2O2 in the cortex. The rationale is that initially dynamic equilibrium exists between the lens and the aqueous humor.

Results

Table B: Results of Model 2 - Necessary GSR to prevent cataract from exceeding LOCS for an amount of time.

LOCS	Crystallin Damage (c.d.)	GSR needed to maintain LOCS for 1 year	GSR needed to maintain LOCS for 20 years	GSR needed to maintain LOCS for 50 years
0.0	0	N/A	N/A	N/A
0.5	0.1327	25.9	N/A	N/A
1	0.2774	23.4	52.49	96.1
1.5	0.4610	21.8	40.79	67.0
2.0	0.6966	20.4	34.78	52.3
2.5	0.9981	18.5	31.19	43.5
3.0	1.3840	16.5	28.83	37.8
4.0	2.5101	10.0	25.90	31.1
5.0	4.3514	0.00	24.05	27.5

Confirmation of Initial Value of GSR

We expect the lens to have some GSR in the lens naturally as part of the antioxidizing system. There will not be enough GSR to fully prevent cataract damage, however. Based on research, if H2O2 levels remain at 10 uM for 1 year, which is common after traumatic accidents affecting the eye, a moderately severe cataract will develop, expected at around the LOCS 4-4.5 level (Fraunfelder 2008).

Literature data estimates that the natural concentration is around 10 uM (Fraunfelder 2008). This corresponds to roughly a LOCS 4 cataract after a year, which confirms that our initial value of GSR in the cortex before any treatment. We will consider this as our starting point, and add GSR to reach this point.

Single Dose

Noyes-Whitney Equation

The Noyes-Whitney equation models the rate of diffusion over time through a diffusion layer, which will be the nanoparticle surrounding the drug, GSR. The following is the Noyes-Whitney equation. If chambers x and y are separated by diffusion layer z, then

$$\frac{dC_x}{dt}=\frac{DS}{V_y h} (C_y - C_x)$$

where Cx is the concentration of the drug in chamber x, Cy is the concentration of the drug in chamber y (assume our drug is 100% soluble), which is across the diffusion layer z from chamber x. Vx is the volume in chamber x, h is the thickness of the diffusion layer z, D is the diffusion coefficient, S is the surface area of the diffusion layer z.

The diffusion constant is defined as follow:

$$D = \frac{k_b T}{6 \pi \mu r}$$

Where $k_b$ is the Boltzmann constant, T is the temperature, $\mu$ is the viscosity of the surrounding solution, and r is the radius of the nanoparticle.

Differential Equations

$$\frac{dC_{in}}{dt}=\frac{DS}{V_{out} h} (C_{out} - C_{in})$$ $$\frac{dC_{out}}{dt}=-\frac{DS}{V_{in} h} (C_{out} - C_{in})$$

From these equations, we get the change in GSR concentration both inside and outside the nanoparticle.

Protein Degradation

All protein degrade over time, and the amount of degradation is proportional to an decreasing exponential function of time. Outside the nanoparticle, protein are degrading. We can apply a decay function on the outside concentration function from the differential equation, to estimate degradation. The turnover rate of the aqueous humor is about 16% per minute, so we expect the degradation factor to be on the order of -5. We define a new function, concentration of GSR outside nanoparticles adjusted for degradation, as a function of time.

$$C_{oa}(t)=C_{out}*e^{-k_dt}$$

Parameters

We know all components of the diffusion constant (temperature, viscosity, natural constants) except for nanoparticle radius. In addition, we find the volume of the aqueous humor by literature data. Other parameters, such as surface area and volume, are determined by the radius.

Determination of Diffusion Layer Dependence

From literature research, we estimate that the diffusion layer is on the order of -11, based on details on turnover rates in the aqueous humor and lens (Toxic). After creating our initial model to guide the experiments, we then used the experimental data to revise our initial model, by altering this constant to -2.5*10^-11

Parameters

Table 2

Variable	Value	Notes	Source
Boltzmann Constant (kb)	$1.3806*10^23 m^2 kg s^{-2}K^{-1}$	Physical Constant
Temperature (T)	$273+37 K$	Body Temperature	Assumed
Viscosity (u)	$1.0*10^{-4} Pa S $		Fluid Mech
Nanoparticle Radius	$200-400 nm$	Analyzed in model	Determined by SEM
Volume of Aqueous Humor (V1)	$2.5*10^{-4} L$	Constant	Medicalopedia
Volume of Nanoparticle (V2)	$10^{-7} L$		Experimental Data from our experiments
Diffusion Layer	(1)$10^{-11} m$ (2)$2.5*10^{-11} m$	Estimate, then revised upon experiments	First estimated with Toxic Data, then revised based on experimental data
Surface Area	$4*\pi*r^2$	Depends on radius
Degradation (kd)	$1.0*10^{-5}$	Estimate, based on turnover rate of lacrimal fluids, in order of -5.	Principles and Practice of Phathalmology Third Edition

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Multiple Dose

Each dose of nanoparticles are released periodically. The function $C_{oa} (t)$ models the GSR increase after a dose given at $t = 0$. If we apply doses periodically, we can apply a time shift to this function by $n*p$, the number of periods multiplied by the period length, to create the graphs shown in Figure C. We also apply a unit step function, so the function value is 0 before the eyedrop is applied By summing up the contribution of each dose, we get the resulting concentration of GSR outside the nanoparticles over time. Remember that our aim is to maintain 43.5 uM of GSR.

$$C_{total}(t)=\sum_{n=0}^\infty {C_{oa} (t-np \cdot)e^-k_d(t-np) \cdot u(t-np)}$$

An example of the resulting $C_{total}(t)$ is shown in Figure D.

Our modeling bares a resemblence to Fourier series, but instead using exponentials to model an

Assumptions

In this model, we made the following assumptions:

1. All nanoparticles are uniform. All nanoparticle doses that are manufactured will be uniform.
2. GSR is delivered directly into the lens.It is difficult to model the function of the lens or diffusion inside the eyes, so we assume almost all GSR that is delivered
3. There is no limit to the concentration of GSR that can be placed into a nanoparticle.This would be an issue of manufacturing, although the concentrations we use are reasonably low to ensure that manufacturing is not a problem.

Analysis of Changing Variables

We vary each variable, nanoparticle radius, concentration, and dose frequency, one at a time, while keeping the other two variables constant.

Changing Radius

We find that changing the radius has little impact on the resulting graph. Equilibrium is slightly decreased when radius is increased. We will ignore nanoparticle radius. We will use whatever range that is most convienient for manufacturers. For our purposes, we will use r = 200 nm.

Although we disregard nanoparticle radius, this simplification is only true in our desired range of 200 to 400 nm. Extremely small or large nanoparticles will show a difference in release.

Changing Concentration

We find that changing the concentration impacts the equilibrium concentration. There is a direct relationship between GSR concentration inside the nanoparticles, and the equilibrium outside. Higher concentrations also make the graph more unstable, however, shown in a greater amplitude of oscillation in Figure 4.C sample graph is shown below.

Changing Frequency

We find that changing the frequency does not impact equilibrium concentration, but does change the stability at equilibrium. The concentration oscillates at a greater amplitude about the equilibrium, which is undesirable, when frequency is too low. Therefore, daily doses are definitely preferred.

We graphed each of the three dependent variables as a function of the three independent variables, and found the quantitative relationship between them. This engineering technique allows us to find an equation for the equilibrium concentration as a function of the three independent variables.

Notice that the time to reach equilibrium (related to the time constant) cannot be altered. See the two eyedrop appraoch to overcome this problem.

Two Step Eyedrop Approach

Explanation

Problem: We cannot decrease the amount of time until the concentration reaches equilibrium, where full prevention is effective.

Solution: We propose the two step eyedrop approach. For the first days of treatment, a higher concentration of eyedrop will be used before using a lower concentration. This will speed up the time to reach equilibrium.

In the diagram, we wish to reach the low equilibrium (in blue) as soon as possible. Using the higher concentrated nanoparticles results in the red curve. By applying the higher concentrated nanoparticles in the first days, then switching to the lower concentrated nanoparticles afterwards, we follow the highlighted path. This allows us to reach equilibrium faster.

We need to figure out the concentration of GSR in naoparticles for the blue and red curves. The blue curve (eyedrop 2) requires the original results of our model (outside collapsible). The red curve (eyedrop 1) requires some calculations.

Finding the Concentration of Eyedrop 1

We cannot change the shape of the concentration graphs, because we cannot change the amount of time to reach equilibrium. This means we can find a formula for the percent of GSR delivered. At equilibrium, the value is 1. By curve fitting an exponential to the graph, we find our "percent delivered" function.

The concentration depends on the desired amount of time to reach equilibriu, which we will call "time limit". By plugging the time limit into our percent function, we will know the percentage delivered at the time limit. We divide our goal concentration by this percentage, to find the equilibrium concentration of the red curve. Using the results our model, we can find the concentration of GSR in nanoparticles of eyedrop 1.

We built a functional calculator to find the concentrations of both eyedrops. This is an extention of the eyedrop provided at the bottom of the Model page.