Team:TAS Taipei/Model

Model - TAS Taipei iGEM Wiki

Model

Cataract prevention occurs over 50 years, so we cannot perform experiments directly on the long-term impact of adding GSR or CH25H. Computational biology allows us to predict cataract development in the long-term. These models allow our team to: (1) understand the impact of adding GSR-loaded nanoparticles into the lens over a 50 year period and (2) design a full treatment plan on how to prevent and treat cataracts with our project. Therefore, the results of our model are essential in developing a functional prototype.

For clarity, we will discuss each model in detail with respect to prevention (using GSR) only. At the end, we extend these results to treatment. In addition, we include collapsibles for interested readers and judges, in order to fully document our modeling work (eg. assumptions, mathematics, and full analysis) while keeping the main page clear with basic points only.

If you are interested in the programming/source codes of these computational models, please contact Avery Wang, at averyw17113532@tas.tw or averyw09521@gmail.com.

Introduction

Guiding Questions

1. How much GSR to maintain in the lens? (GSR Function)

2. How to maintain that amount of GSR using nanoparticles and eyedrops? (Delivery Prototype)

Focus of Models

Since our construct is not directly placed into the eyes, how our synthesized protein impacts the eye after it is separately transported into the lens is of greater importance. As a result, we create models with the intent on understanding how GSR and CH25H impacts the eye, and how we can control its impact with a well-designed delivery prototype.

Prevention: GSR Function

Model 1: Crystallin Damage

The amount of damage to crystallin by H2O2 determines the severity of a cataract (Spector 1993). We relate the amount of crystallin damage to the corresponding rating on the LOCS scale, used by physicians to rate cataract severity. Our goal is to lower LOCS to below 2.5, the threshold for surgery. Through literature research as well as our own experimental data, we find the maximum allowable crystallin damage to prevent a LOCS 2.5 cataract from developing.

Measurement of Cataract Severity

There are three ways of measuring cataract severity, each used for a different purpose.

1. Lens Optical Cataract Scale (LOCS): Physicians use this scale, from 0 – 6, to grade the severity of cataracts (Domínguez-Vicent 2016).
2. Absorbance at 397.5 nm: This is the experimental method, used by our team in the lab (c.d.)
3. Crystallin Damage: This is a chemical definition. We quantify cataract severity as a function of how much oxidizing agents there are, as well as how long crystallin is exposed to oxidizing agents. (Cui X.L. 1993)
When we add GSR into the system, crystallin damage decreases (Cui X.L. 1993). Our goal is to find the equivalent crystallin damage that leads to a LOCS 2.5 cataract, in order to know how much GSR to prevent this amount of crystallin damage in Model 2.

LOCS to Absorbance: Literature Data

Numerous studies show how absorbance measurements can be converted to the LOC scale that physicians use. With the results of Chylack, we construct the first two columns in Table 1 (Chylack 1993).

Absorbance Equivalence to Crystallin Damage: Experimental Data

We use experimental measurements from our team’s Cataract Lens Model (Link). They induced an amount of crystallin damage, and measured the resulting absorbance. The experimental results are listed in Table A, and Figure 4.A, we calculate the equivalent crystallin damage of each LOCS rating and absorbance, and create the third column of Table 1.

Table 1: Results of Model 1 – Equivalent values for LOCS, Absorbance, and Crystallin Damage. First two columns come from Chylack, the third column uses data from our team's Lens Model experiment (see collapsible)
LOCS Absorbance (@397.5 nm
abs units)
Crystallin Damage
(M-h)
0.0 0.0000 0.0000
0.5 0.0143 0.1243
1.0 0.0299 0.2878
1.5 0.0497 0.4697
2.0 0.0751 0.6949
2.5 0.1076 0.9883
3.0 0.1492 1.3747
4.0 0.2706 2.5259
5.0 0.4691 4.3472

Conclusion

To guarantee that surgery is not needed for 50 years, we need to limit crystallin damage to 0.9883 units. If crystallin damage goes above this threshold, then surgery is needed. This is the crystallin damage threshold for a LOCS 2.5 cataract.

Motivation

Crystallin damage is the chemically significant way to quantify cataract severity (Cui X.L. 1993). Physicians grade cataracts on a LOCS scale from 0 to 6, with 2.5 usually being the threshold for surgery. This standard method to determine cataract severity is through optical analysis, which is not applicable for chemical analysis. (Chylack)

The purpose of this model is simple. In Model 2, we find the amount of GSR to maintain in order to ensure a cataract remains below a certain severity. Before we can do that, we must find a way to relate the physical definition of cataract severity (LOCS) to the chemical definition (crystallin damage) in Model 1.

We use literature data to relate LOCS to absorbance, which is still a physical property, but is frequently used in the lab. Then, with experimental data, we relate crystallin damage to absorbance, thus completing the link between the chemical and the physical definitions.

Quantification of Crystallin Damage

Crystallin damage is difficult to quantify, as few literature sources attempt to quantify it. We propose a mathematical way to measure crystallin damage based on how complete the degradation reaction of crystallin is complete depending on the amount of H2O2.

Crystallin damage depends on two factors: (1) the amount of H2O2 crystallin is exposed to, and (2) the time crystallin is exposed to H2O2. Therefore, we can mathematically quantify this by integrating the amount of H2O2 in the crystallin over time: $c.d.(t) = \int_{0}^{\infty} [H_2O_2]_t dt$

We define one unit of crystallin damage to be the equivalent damage caused by exposing the amount of crystallin in the human eyes 1 M of H2O2 for 1 hour, with units M-h. We make an assumption here, described fully in “Assumptions” #1.

Note that 1 M-h is a significant amount of crystallin damage, because the concentration of crystallin in the eyes is extremely small, in the neighborhood of 10 uM (Fraunfelder 2008). In addition, the eyes have a naturally occurring antioxidant system that lowers the concentration of H2O2, so it will take much longer than an hour of exposure until 1 M-h of crystallin damage is reached.

Data Documentation

LOCS to Absorbance

The results of Chylack et. Al. relate LOCS to opacity. Opacity (%) is physically related to absorbance at 397.5 nm, and can be calculated with the following equation.

As a result, we can find the equivalent absorbance of each point on the LOCS scale, shown in Table 1 (outside collapsible).

Absorbance to Crystallin Damage

 Trial H2O2 Concentration (M) Exposure Time (h) Crystallin Damage (c.d.) Measured Absorbance (abs @400 nm) 1 0.100 24.0 2.40 0.105 2 0.100 46.5 4.65 0.451 3 0.100 72.0 7.20 0.695 4 0.100 20.0 2.00 0.089 5 0.100 42.0 4.20 0.392 6 0.100 15.0 1.50 0.093 7 0.100 42.0 0.340 0.340 8 0.100 67.0 6.70 0.563

Assumptions of Model 1

We are about to use experimental data from experiments we performed. As we cannot perfectly simulate a human cataract in our lab, we have to make the following assumptions:

1. Cataract damage is directly proportional to H2O2 concentration and time. Crystallin damage is mainly caused by H2O2 interacting the cysteine on crystallin. The reaction between H2O2 and cysteine is first-order (Domínguez-Vicent 2016), supporting this assumption. This allows us to make the quantification of crystallin damage.
2. We assume that the amount of crystallin in the lens is far greater than the amount of crystallin degraded. Therefore, a change in the amount of crystallin does not impact crystallin damage, as H2O2 only degrades a small portion of crystallin that causes blurred lens. (Cui X.L 1993)
3. The crystallin in humans is similar in function as crystallin in fish. This allows us to use experimental data on fish into this model.
4. When the experiments diluted the cataract lens protein, the amount of crystallin is diluted. However, the final absorbance of degraded crystallin is also diluted, so we assume any errors in absorbance is canceled out.
5. In the fish used, we assume there is there is no GSR. GSR decreases the amount of H2O2 in the fish, but since we could not find the amount of GSR in the fish, we must make this assumption.

Using Experimental Data for Model

In each trial (see Data Documentation), we added H2O2 of a known concentration to fish lens, and allowed crystallin to degrade for some set time. With all four assumptions, we can calculate the theoretical crystallin damage that occurred. Meanwhile, experimentally we found the absorbance of the degraded crystallin. We find a link between crystallin damage and absorbance.

When determining the relationship between absorbance and crystallin, in Figure 4.A the best fit line has a x – intercept that is nonzero. However, when converting each absorbance rating to equivalent crystallin damage in Table 1, we ignore the constant term. When doing the experiments, the fish lens may have contained GSH that is still active, so the fact that the crystallin is exposed to H2O2, the degradation reaction does not happen until all GSH is depleted, and crystallin damage begins to form. We subtract around 1 unit of crystallin damage from all values, as shown in Figure 4.B.

Error Analysis

There are some limitations of the model that arise from our assumptions. We assume that fish and human lens contain similar crystallin proteins that are degraded in the similar manner. In addition, we made a rough adjustment of data based our diluting procedure. For better results to create a human cataract model, experiments will need to be done on human lens, even better if done in vitro, without any dilutions.

Significance of Model

With the results of this experiment, we create Table 1, which equates each LOCS value with the equivalent absorbance.When we experiment with the cataract lens model, we measure the absorbance. However, the absorbance values are meaningless to physicians and patients. This model allows them to translate absorbance into LOCS, which tells them exactly whether they need surgery, or into crystallin damage, which tells them how much GSR they need.

Model 2: GSR Pathway

Now that we know how much we need to limit crystallin damage to LOCS 2.5, we model the naturally occurring GSR Pathway in the lens of a human eye. We calculate the necessary GSR concentration to be maintained over 50 years so that the resulting cataract is below LOCS 2.5.

Chemical Kinetics Model: Differential Equations

By various enzyme kinetics laws, fully documented in the collapsible, we build a system of 10 differential equations based on 6 chemical reactions (Ault 1974, Pi 2004). All parameters, constants (Table B), and initial conditions (Table C) are based off literature data (Ng 2007, Melissa 2012, Saravanakumar 2015, Salvador 2005, Adimora 2010, Jones 2008, Martinovich 2005). Estimates made are also shown with assumptions and reasoning. The details are shown in the collapsible for interested readers.

Blackbox Approach: Testing GSR Impact

Our modeling approach is seen in Figure 4.2, where vary the input, Initial GSR concentration, holding all other variables constant, and numerically solve for the amount of hydrogen peroxide over time. We can find the amount of crystallin damage accumulated over 50 years if different levels of GSR is maintained, which we graph in Figure 4.3.

From this graph, we can find the GSR concentration needed for the LOCS 2.5 threshold.

Crystallin Damage vs. GSR Level

According to literature data and our model, the naturally occurring GSR concentration is 10 uM (Fraunfelder 2008). All curves show crystallin damage decreasing as GSR levels are increased, which supports both research and experimental data, and suggests that this prototype is effective in preventing crystallin damage. However, GSR levels need to be raised significantly, up to 40+ uM from the natural 10 uM of GSR in order to show long-term protection.

Figure 4.3 shows the amount of GSR we need to maintain for 50 years in order to prevent a LOCS cataract of a certain severity. The row of interest is LOCS 2.5, the threshold for surgery. Notice that we say “maintain” the level of GSR. This level needs to be constant at all times for 50 years for full prevention. The delivery of GSR to maintain this level is discussed in Model 3.

Conclusion

We need to maintain (NOT add) 43.5 uM of GSR in the lens so that the crystallin damage recorded over 50 years is below the LOCS 2.5 threshold.

Introduction

We have already found the threshold for crystallin damage, in a unit that is related to hydrogen peroxide concentration. Now, we find how much GSR would limit hydrogen peroxide enough so crystallin damage falls below the threshold.

Because of our definition of crystallin damage, by integrating a hydrogen peroxide vs. time graph, we effectively calculate crystallin damage. We build our model, with the goal of finding the hydrogen peroxide over a period of 50 years.

We chose 50 years, because we assume prevention would start at an early age, such as the 30s to 40s, when cataracts may start developing. Then, our prototype will prevent cataracts for one’s entire lifetime.

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.

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

Parameters & Initial Conditions

Table B: Parameters for Model 2, and their sources/estimates.
VariableValue Notes Source
k1 $2.1\cdot10^7$ Rate constant for Reaction 1 (Ng 2007)
k2 $4\cdot10^4$ Rate constant for Reaction 2 (Ng 2007)
k3 $1\cdot10^7$ Rate constant for Reaction 3 (Ng 2007)
K4m $0.063\cdot10^{-3}$ Michaelis constant for NADPH (Salvador 2005)
K5m $0.154\cdot10^{-3}$ Michaelis constant for GSR (Saravanakumar 2015)
K4c $0.219\cdot10^{-3}$ Michaelis constant for NADPH (Salvador 2005)
K5c $7.985\cdot10^{-3}$ Michaelis constant for GSR (Saravanakumar 2015)
K6 $10^{-5}$ Diffusion constant Estimate, derived from (Adimora 2010,) estimate
Table C: Initial Conditions of Model 2
ChemmicalInitial Condition (M) Source
H2O2 (in) $0.5\cdot10^{-3}$ (Adimora 2010)
H2O2 (out) $0.5\cdot10^{-3}$ Estimate, assuming H2O2 in and otu are in dynamic equilibrium.
GPxr $5\cdot10^{-5}$ (Ng 2007)
GPxo $10^{-14}$ (Ng 2007)
H2O $0$ Ignored, does not impact system
GSH $3.68\cdot10^{-5}$ (Jones 2005)
GSGPx $10^{-14}$ Estimate, similar to GPxo
GSSG $1.78\cdot10^{-6}$ (Ng 2007)
NADPH $3.0\cdot10^{-3}$ (Martinovich 2005)
GSR1 Varied We change this as an input in our Model
GSR2 $0$ Assuming no reaction has taken place yet.

Procedure

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 4.3).

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 D: 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 (uM) GSR needed to maintain LOCS for 20 years (uM) GSR needed to maintain LOCS for 50 years (uM)
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 0.9883 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.

Prevention: Prototype Function

Model 3: Nanoparticle Protein Delivery

To maximize delivery efficiency to the lens, we encapsulate GSR in chitosan nanoparticles (Wang 2011, Tajmir-Riahi 2014). From Models 1-2, we have found the necessary concentration of GSR that needs to be maintained in the lens. Now we design nanoparticles that will maintain those amounts. We build a model find how nanoparticles release GSR at appropriate rates to control the amount of GSR in the lens, and find the best engineered design for nanoparticles.

Single Dose: Change in GSR Concentration

In finding the best engineered design, we take into account variables such as nanoparticle radius and concentration. We build a differential equation model for the impact of a single dose of nanoparticles over time. To generalize the model, instead of using absolute concentrations, we use relative concentration, with respect to the natural amount, or initial amount of GSR in the lens. The full mathematics and details can be found in the collapsible.

We get two curves, the concentration of GSR outside nanoparticles subjected to degradation (Figure 4.4), and the release of GSR from nanoparticles (Figure 4.5), over time. This allows us to predict nanoparticle delivery rates before we perform the actual experiments  (Link)

Comparison with Experimental Data

Yet in our model, we do not know the thickness of the nanoparticle diffusion layer. After performing experiments, we can use measurements of our prototype device to find this thickness, and refine our model. A direct comparison of our model with our experiment data is shown in Figure 4.5.

Multiple Dose: Change in GSR Concentration

Each dose of nanoparticles, represented in the Single Dose model, can be repeated to create the Multiple Dose model. Below is a graph of GSR concentration over time when multiple doses of nanoparticles are added.

In Figure 4.6, all curves approach equilibrium, after which the concentration oscillates about equilibrium. We have three goals, in order of importance for best nanoparticle design:

1. GSR equilibrium concentration equal to amount we desire (i.e. 43.5 uM from Model 2)
2. Stability of concentration at equilibrium (Model 4 goes into deeper depth regarding sensitivity)
3. Time to reach equilibrium (time for full prevention to come into effect)

To do so, we can alter different variables: GSR concentration in nanoparticles, nanoparticle radius, and dose frequency. For a full analysis of how each variable impacts the concentration function, see the collapsible. Below is a summary of the results:

Table 2: Results of Analysis, where we varied one independent variable at a time, and keeping the other two constant. The effect on the dependent variable is described in the table.
Independent Variable Time to Reach Equilibrium Equilibrium Concentration Stability
Concentration No impact Proportional Increase Slight Decrease
Radius No impact Very Slight Decrease Very Slight Decrease
Dose Frequency No impact No impact Increase

We find the optimal combination of parameters is daily doses (high frequency) of 200 nm nanoparticles (small), with a concentration of 76.88 uM of GSR in the nanoparticles (concentration).

The calculator at the end of the page can be altered, so if the LOCS threshold is not 2.5, a new concentration can be calculated.

A Two Stage Eyedrop Approach

As shown in Table 2, we cannot alter the time to reach equilibrium, or reach full prevention. As supported by literature research, the time to reach equilibrium is a property of the lens that we cannot change (Fraunfelder 2008). However, we propose a two-stage eyedrop approach, of two differing nanoparticle concentrations, to decrease the time needed for full prevention. A full explanation is found in the collapsible.

Generalized Nanoparticles: Customizer

We built a full nanoparticle customizer, which generalizes the model to beyond delivery into the eye, found at the end of the page (Software). We hope that other iGEM teams who are interested in nanoparticle drug delivery can utilize this customizer to help them develop their own prototype.

Conclusion

We find the optimal combination of parameters is daily doses (high frequency) of 200 nm nanoparticles (small), with a concentration of 76.88 uM of GSR in the nanoparticles (concentration).

Introduction

We do not put GSR directly into eyedrops, because the high turnover rate of the aqueous humor results in a lot of GSR being washed away. Instead, we encapsulate GSR into chitosan nanoparticles, which results in a greater amount of GSR delivered to the lens. This model will quantify the amount of GSR delivered to the lens over time, as new eyedrops are periodically applied.

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 (Fraunfelder 2008).

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.

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.

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

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

For initial conditions, the concentration inside the nanoparticles start at an initial value, while the concentration outside the nanoparticles start at the natural concentration of GSR (see Model 2 collapsible).

Parameters

 Variable Value Notes Source Boltzmann Constant (kb) $1.3806 \cdot 10^{23} \cdot m^2 \cdot kg \cdot s^{-2} \cdot K^{-1}$ Physical Constant Temperature (T) $273+37 K$ Body Temperature Assumed Viscosity (u) $1.0 \cdot 10^{-4} Pa \cdot S$ Medicalopedia Nanoparticle Radius $200-400 nm$ Analyzed in model Determined by SEM Volume of Aqueous Humor (V1) $2.5\cdot 10^{-4} L$ Constant Medicalopedia Volume of Nanoparticle (V2) $10^{-7} L$ Estimate with Experimental Data from our experiments Diffusion Layer (1)$10^{-11} m$(2)$2.5\cdot10^{-11} m$ Estimate, then revised upon experiments (Fraunfelder 2008) Surface Area $4 \pi r^2$ Depends on radius Degradation (kd) $1.0\cdot10^{-5}$ Estimate, based on turnover rate of lacrimal fluids, in order of -5. (Fraunfelder 2008)

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 $n\cdot p$, 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 equation bares a resemblence to Fourier series, but instead we use exponentials to model the concentration function. The analysis we take later is similar to how Fourier series are analyzed.

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.

We find that changing the radius has little impact on the resulting graph, shown in Figure 4.D. 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.E sample graph is shown above.

Changing Frequency

We find that changing the frequency does not impact equilibrium concentration, but does change the stability at equilibrium, as shown in Figure 4.F. 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 Stage 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 stage 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.

Prevention

LOCS Score Threshold:
Time Limit (Days)
We guarentee that by applying this prevention eyedrop daily, your LOCS score will remain below your threshold for 50 years.

Variable Eyedrop 1 Eyedrop 2
Nanoparticle Conc.      uM      uM
Eyedrop Result      mg/mL      mg/mL

Model 4: Eyedrop Prototype

We have found a nanoparticle design to deliver GSR. We also need to model the function of eyedrops, to determine the concentration of GSR-loaded nanoparticles to put in eyedrops, and analyze how sensitive the resulting system is.

Bioavailability of GSR Delivery

The eye is well protected from foreign material attempting to enter the eye. The corneal epithelium is the most essential barrier against topical drugs in eyedrops, and as a result, much of drugs in eyedrops are lost in tear drainage (Lux 2003).

Bioavailability describes the proportion of the drug that reaches the site of action, regardless of the route of administration. For example, it is estimated that only 1-5% of an active drug with small solutes in an eyedrop penetrates the cornea (Bonate 2005). In the case of nanoparticles, which are much larger than chemical molecules, more is lost (Fraunfelder 2008).

The results show that the bioavailability of nanoparticles is about 1.404 x 10 -3%, which means that for every gram of GSR (or any drug) we place into nanoparticles, approximately 14.04 ug of the drug reach the aqueous humor. The variance is 2.34 ug/g. (Calvo 1996)

To ensure that sufficient concentrations of GSR are delivered, we must place an excess of GSR. To determine how much, we simply divide the concentration of GSR in nanoparticles we found in Model 3 by the fraction of GSR that reaches the aqueous humor. This result is programmed in our calculator below.

We conclude that we need 5.48 mM of GSR in nanoparticles in our final eyedrop to maintain 43.5 uM GSR and thus 2.5 LOCS.

Sensitivity Analysis: Revisiting Nanoparticles Model

The mechanism for eyedrop delivery is complex, and there are variances in the bioavailability depending on the conditions of the eye (Fraunfelder 2008). The thickness of the cornea, lens, other eye diseases, age, and even time of day may impact the bioavailability of the drug (Gaudana 2010). We use a stochastic model to simulate Model 3 again, but this time, add a degree of variance. The result is shown in Figure 4.10.

The variance is impacted by the frequency of eyedrops. By giving eyedrops more frequently with less amounts given each time, the variance is decreased.

Ideally, we wish to deliver 100% of the GSR concentration of the amount found in Model 2 (43.5 uM). Because of variance, the actual amount maintained in the lens is different, shown in Figure 5. The full details and mathematics of the stochastic model can be found in the collapsible.

Stochastic Analysis

Mathematica provides a built-in process for stochastic analysis. We try to model our percent function (from Model 3), which calculates the ratio of current GSR versus final equilibrium GSR. Our percent function is: $$P(t)=1-1.027*e^{-0.1717t}$$ where t is in days.

Our stochastic model is based off the following differential equation, written in differential form: $$dC_R=dP + \sigma \frac{dP}{dt} dw(t)$$ where $C_R$ is the concentration ratio of current to final at equilibrium (like $P$, ranges from 0 to 1), $P$ is the percent function, $\sigma$ is the standard deviation of delivery, and $w(t)$ approximates the stochastic Wiener stochastic process.

The standard deviation is found from the square root of the variance divided by the equilibrium to get a number between 0 and 1, multiplied by a factor related to the frequency. The greater the frequency, the less GSR there is to every dose of eyedrop, so the standard deviation would decrease, and vice versa.

For full details, please see the research paper: Hanson, F.B. (2007).Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation. Chicago, IL:The Society for Industrial and Applied Mathematics.

Insights into Manufacturing & Clinical Use

Using the details from Models 1-4, we offer the following suggestions for manufacturing our eyedrops, as well as prescribing them to patients.

1. Frequent doses of low concentration eyedrops are more stable than occasional doses of high concentration eyedrops. (Model 3)
2. Cataract severity is extremely sensitive to the amount of GSR concentration maintained. If this value falls below 43.5 uM even by a little, as shown in Model 2, cataracts may develop. (Model 2)
3. When manufacturing, there should be a necessary upward adjustment, because even with daily eyedrops, some simulated trials resulted in less than 90% of GSR delivered. (Model 4)
4. Cataract prevention is only fully effective after around 28 days of using the prevention eyedrop. (Model 3) During that time, further cataract damage may take place. The suggested order of treatment should be: start with GSR eyedrops until full prevention is in effect, the use 25HC eyedrops to reverse any existing cataract. Finally, stop 25HC eyedrop use, while continuing GSR eyedrops. (Model 1)
5. To speed up the time for full prevention to take place, a two eyedrop approach to quickly deliver sufficient GSR should be used. (Model 4)

Treatment

We use the results from our previous models, and apply them to treatment. The process is almost the exact same as that of Prevention. The only difference is that our treatment protein, CH25H, reverses cataract damage (Griffiths 2016). We find the exact concentration of CH25H to reverse a cataract of a given LOCS score. The delivery models are unchanged, with the exception that the concentration of protein delivery will be different.

We use the results of Model 1 to calculate the LOCS equivalent crystallin damage we need to reverse (“negative crystallin damage”). Then we use experimental results to calculate the concentration of CH25H needed to reverse the cataract. We do not use Model 3, as this is a one-time treatment. After applying the results of Model 4, we can find the final concentration needed in CH25H eyedrops.

We propose eyedrops with 0.8 mg/mL CH25H. The number of drops needed for treatment depends on a patient's current LOCS score, and is calculated in the software tool below.

CALCULATOR

Prevention

LOCS Score Threshold:
We guarentee that by applying this prevention eyedrop daily, your LOCS score will remain below your threshold for 50 years.

Prevention Results
Variable Value Source
Allowable LOCS
Crystallin Damage       c.d. Model 1
GSR Maintained      uM Model 2
Nanoparticle Conc.      uM Model 3
Eyedrop Conc.      mM Model 4
Eyedrop Result      mg/mL

Treatment

By applying the following treatment, leaving an hour before each dose of eyedrops, we guarentee that it will lower your LOCS score to essentially 0.

Treatment Results
Variable Value Source
Allowable LOCS
Crystallin Damage       c.d. Model 2
Absorbance      a.u. Model 1
CH25H      uM Model 5
Eyedrop Conc.      uM Model 4
Eyedrop Result      mg/mL Model 4
# of Eyedrops      drops (of 0.8 mg/mL eyedrop)

Software

We built a nanoparticle customizer, which allows you to track the concentration and rates of drug delivery to any part of the body using nanoparticles. You can customize your own design of nanoparticles, and analyze its function inside the body.

This computational software can be used by future iGEM teams who are interested in using nanoparticles to efficiently deliver their synthesized proteins. The calculational tool is programmed on a Google Spreadsheet. Click the button below to visit the spreadsheet. Please make a copy of the spreadsheet to freely use it.

Conclusion

The models allowed us to extrapolate short-term experimental data to the long-term, to over 50 years, the duration for cataracts to form. In Models 1-2, with the Cataract Lens Experiments on our synthesized protein GSR (and CH25H), we found the amount of synthesized protein needed in the lens over 50 years. In Models 3-4, with the experimental results of a single dose of nanoparticle release, we simulate multiple doses to see the overall change in GSR concentration in the lens over 50 years. We calculate the exact concentrations of our eyedrop prototype, to prevent a LOCS 2.5 cataract from developing so surgery is not needed. With our predictions and simulations of cataract treatment, we offered insights into how the product should be manufactured and used in clinics. We also programmed a nanoparticle customizer to allow the results of these models to be generalized to all drug delivery iGEM projects.

Citation

Adimora, N.J., Jones, D.P., & Kemp, M.L. (2010). A model of redox kinetics implicates the thiol proteome in cellular hydrogen peroxide responses. Antioxidants & Redox Signaling, 13(6), 731-43

Ault, A. (1974). An introduction to enzyme kinetics. Journal of Chemical Education,51(6), 381, DOI: 10.1021/ed051p381

Bonate, P.L., & Howard, D.R. (2005) Pharmacokinetics in Drug Development Volume 2: Regulatory and Development Paradigms. American Association of Pharmaceutical Scientists

CALVO, P., ALONSO, M. J., VILA‐JATO, J. L., & ROBINSON, J. R. (1996). Improved ocular bioavailability of indomethacin by novel ocular drug carriers. Journal of Pharmacy and Pharmacology, 48(11), 1147-1152.

Chylack, L.T., Wolfe, J.K., Singer, D, M., Leske, C., Bullimore, M, A., Bailey, I.L., Friend, J., McCarthy, D., & Wu, S. (1993). The Lens Opacities Classification System III. The Longitudinal Study of Cataract Study Group. American Medical Association. 111(6), 831-36.

Cui, X. L., & Lou, M. F. (1993). The effect and recovery of long-term H 2 O 2 exposure on lens morphology and biochemistry. Experimental eye research, 57(2), 157-167.

Dominguez-Vincent, A., Birkeldh, U., Carl-Gustaf, L., Nilson, M., Brautaset, R., & Al-Ghoul. K.J. (2016).Objective Assessment of Nuclear and Cortical Cataracts through Scheimpflug Images: Agreement with the LOCS III Scale. PLoS One.11(2), doi: 10.1371/journal.pone.0149249

Fraunfelder, F.T., Fraunfelder F.W., & Chambers, W.A.(2008) Clinical Ocular Toxicology: Drug-Induced Ocular Side Effects. Saunders

Gaudana, R., Ananthula, H.K., Parenky, A., Mitra, A.K. (2010). Ocular Drug Delivery. AAPS Journals, 12(3), 348-360 doi: 10.1208/s12248-010-9183-3

Griffiths, W.J., Khalik, J.A., Crick, P.J., Yutuc, E., & Wang, Y. (2016). New methods for analysis of oxysterols and related compounds by LC–MS. Journal of Steroid Biochemistry and Molecular Biology, 162, 4-26, http://dx.doi.ordg/10.1016/j.jsbmb.2015.11.017

Jones, D.P. (2008). Radical-free biology of oxidative stress. American Journal of Physiology. Cell Physiology, 295(4), 849-68

Lux, A., Maier, S., Dinslage, S., Suverkrup, R., & Diestelhorst, M.(2003). A comparative bioavailability study of three conventional eye drops versus a single lyophilisate. British Journal of Ophthalmology, 87(4), 436-440.

Martinovich, G.G., Cherenkevich, & S.N., Sauer. (2005). Intracellular redox state: towards quantitative description. European Biophysics Journal, 34(7), 937-42

Ng, C.F., Schafer, F.Q., Buettner, G.R., & Rodger, V.G.(2007).The rate of cellular hydrogen peroxide removal shows dependency on GSH: mathematical insight into in vivo H2O2 and GPx concentrations. Free Radical Research, 41(11), 1201-11

Norlin, M., Andersson, U., Bjorkhem, I., & Wikvall, Kjell. (2000). Oxysterol 7α-Hydroxylase Activity by Cholesterol 7α-Hydroxylase (CYP7A)*. The American Society for Biochemistry and Molecular Biology, J. Biol. Chem. 34046-34053. doi:10.1074/jbc.M002663200

Salvador, A., Savageau, M.A. (2005). Evolution of enzymes in a series is driven by dissimilar functional demands. National Academy of Sciences of the USA, 103(7), 2226-2231.

Saravanakumar, S., Eswari, A., & Rajendran, L. (2015). Mathematical analysis of immobilized enzyme with reaction-generated pH change on the kinetics using Asymptotic methods.International Journal of Advanced Multidisciplinary Research, 2(8), 98-119

Spector, A., Wang, G.M., Wang, R.R., Garner, W.H., & Moll, H. (1993). The prevention of cataract caused by oxidative stress in cultured rat lenses. I. H2O2 and photochemically induced cataract.Current Eye Research, 12(2), 163-79

Pi, N. A., Yu, Y., Mougous, J. D., & Leary, J. A. (2004). Observation of a hybrid random ping‐pong mechanism of catalysis for NodST: A mass spectrometry approach. Protein science, 13(4), 903-912.

Tajmir-Riahi, H.A., Nafisi, Sh., Sanyakamdhorn, S., Agudelo, D., & Chanphai, P. (2014). Applications of chitosan nanoparticles in drug delivery. Methods in Molecular Biology, 165-84, doi: 10.1007/978-1-4939-0363-4_11

Torchilin, V.P.(2006). Nanoparticles as Drug Carriers. Imperial College Press

Wang, J.J., Zeng, Z.W., Xiao, R.Z., Xie, T., Zhou, G.L., Zhan, X.R., & Wang, S.L. (2011). Recent advances of chitosan nanoparticles as drug carriers. Int J Nanomedicine, 6, 765-774 doi: 10.2147/IJN.S17296

Wong, M.M., Shukla, A.N., & Munir, W.M.(2012). Correlation of corneal thickness and volume with intraoperative phacoemulsification parameters using Scheimpflug imaging and optical coherence tomography. Journal of Cataract & Refractive Surgery, 40(12), 2067-75 doi: 10.1089/ars.2009.2968.

Eyedrops

× Zoom out to see animation.
Your screen resolution is too low unless you zoom out