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 sake of 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.
Introduction
Guiding Questions
How much GSR to maintain in the lens? (GSR Function)
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. 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.
Lens Optical Cataract Scale (LOCS)>: Physicians use this scale, from 0 – 6, to grade the severity of cataracts.
Absorbance at 397.5 nm: This is the experimental method, used by our team in the lab (c.d.)
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.
When we add GSR into the system, crystallin damage decreases. 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.
Crystallin damage is the chemically significant way quantifying cataract severity.
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.
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 rarely measured chemically. 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. The chemical equation is as follow:
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. 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
Data Documentation
LOCS to Absorbance
The results of Domínguez-Vicent 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
Table A: Experimental Data used for Model 1 from Cataract Lens Model (TAS) - Absorbance vs. 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
0.0.695
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
Assumptions of Model
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:
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,), supporting this assumption. This allows us to make the quantification of crystallin damage.
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.
The crystallin in humans is similar in function as crystallin in fish. This allows us to use experimental data on fish into this model.
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.
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.
Data Analysis
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.
Model Adjustment
When determining the relationship between absorbance and crystallin, in Figure 1 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.
Analysis
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.
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 ________ and ________, we construct the first two columns in Table 1.
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.
With this relation graphed in Figure 2, 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, Opacity, Absorbance, and Crystallin Damage.
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.
Figure 2: Cataract Threshold
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. All parameters, constants, and initial conditions are based off literature data. Estimates made are also shown with assumptions and reasoning. The details are shown in the collapsible for interested readers.
Blackbox Approach: Testing GSR Impact
We 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 2.
From this graph, we can find the GSR concentration needed for the LOCS 2.5 threshold.
Figure 2: Cataract Threshold
Crystallin Damage vs. GSR Level
According to literature data and our model, the naturally occurring GSR concentration is 10 uM. 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 2 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.
Each reaction will be discussed in detail, and we will derive rate equations.
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.
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:
Crystallin damage is linear with respect to both concentration and time, as evidenced by cysteine’s first order reaction with hydrogen peroxide (Domínguez-Vicent). (see Model 1 for full explanation).
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 100 days of the treatment have passed. As we will discuss in Model 3, a one-time treatment will be made as soon as prevention is fully effective.
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.
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.
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.
Literature data estimates that the natural concentration is around 10uM. 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.
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.
Prevention: Prototype Function
Model 3: Nanoparticle Protein Delivery
To maximize delivery efficiency to the lens, we encapsulate GSR in chitosan nanoparticles. 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, concentration of GSR in the nanoparticles, and GSR release from nanoparticles, over time. This allows us to predict nanoparticle delivery rates before we perform the actual experiments.
Figure 3. Single Dose of Nanoparticles Model of how GSR concentration in the lens is increased as a result of one dose of nanoparticles releasing GSR. Each line represents a different concentration of GSR in the nanoparticle. The GSR concentration in the lens increases, then decreases back to the initial, due to degradation.
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 ___.
Figure 4. Nanoparticle Release: Experiments vs. ModelThis is a graph of the amount of GSR left in nanoparticles as a function of time. As GSR is released, the amount of GSR in the nanoparticle falls as a decay exponential. We create a model to guide our experiments, and then use experimental data to refine our model.
Multiple Dose: Change in GSR Concentration
In Figure ____, all curves approach equilibrium, after which the concentration oscillates about equilibrium. We have three goals, in order of importance for best nanoparticle design:
GSR equilibrium concentration equal to amount we desire (i.e. 43.5 uM from Model 2)
Stability of concentration at equilibrium (Model 4 goes into deeper depth regarding sensitivity)
Time to reach equilibrium (time for full prevention to come into effect)
Figure 4. Nanoparticle Release: Experiments vs. Model
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
Independent Variable
Time to Reach Equilibrium
Equilibrium Concentration
Stability
Concentration
No impact
Proportional Increase
Slight Increase
Radius
No impact
Slight Decrease
No impact
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.
Figure 5: Something
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 ______, the time to reach equilibrium is a property of the lens that we cannot change. However, we propose a two-step eyedrop approach, of two differing nanoparticle concentrations, to decrease the time needed for full prevention. A full explanation is found in the collapsible.
Figure 5: Something
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).
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.
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 (Schoenwald 1997). In the case of nanoparticles, which are much larger than chemical molecules, more is lost. (Clinical Ocular Toxicology)
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)
Figure 6: Something
Necessary Adjustments in Eyedrops
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.
[Calculations]
We conclude that we need 5.48 mM of GSR in nanoparticles 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. The thickness of the cornea, lens, other eye diseases, age, and even time of day may impact the bioavailability of the drug. We use a stochastic model to simulate Model 3 again, but this time, add a degree of variance.
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.
Treatment
We use the results from our previous models, and apply them to treatment. The only difference is that our treatment protein, CH25H, reverses treatment. 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 is calculated in the software below.
Figure 7: Something
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
LOCS Score Threshold: 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)
Conclusion
Prevention
GSR Eyedrop
Treatment
25HC Eyedrop
LOCS: 0
Eyedrops
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