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                                        <p> The key question: <b>How much GSR to add?</b> Now that we know how much we need to limit crystallin damage, we use systems of ordinary differential equation to model the GSR Pathway. We calculate the necessary GSR concentration to be maintained over 50 years so that the resulting cataract is below LOCS 2.5./p>
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                                        <p> How much GSR do we need to maintain in the lens so that the crystallin damage recorded over 50 years is below the threshold for LOCS 2.5? </p>
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                                        <p> By the Law of Mass Action, Michaelis-Menten Enzyme kinetics, Ping-pong mechanism, and the Law of Passive Diffusion, 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. </p>
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                                        <h3>Blackbox Approach: Testing GSR Impact </h3>
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                                        <p> We will vary the input, Initial GSR concentration, from 0 to 100 uM, holding all other variables constant, and numerically solve for the amount of hydrogen peroxide over time. With this graph, we can find the amount of crystallin damage accumulated over 20 to 50 years if different levels of GSR is maintained.  </p>
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                                        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 2, 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.
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Revision as of 13:22, 30 September 2016

Modeling - TAS Taipei iGEM Wiki




C◎UNTERACTS

Modeling

Abstract

We answer two questions: How much GSR to maintain in the lens, and how to maintain that amount? We find the amount of GSR needed in the lens (Model 2) to limit crystallin damage so the resulting cataract is less than LOCS 2.5 (Model 1). Then, we find the optimal design of eyedrops (Model 4) and nanoparticles that will maintain this amount of GSR in the lens (Model 3). These models allow our team to understand the impact of adding GSR-loaded nanoparticles into the lens, and to design a full treatment plan on how to prevent and treat cataracts.

Achievements

  • Designed a simple calculator to find amount of GSR or 25HC eyedrops needed for a patient's LOCS score.
  • Bridged the gap between the medical, biological, and chemical measurement of crystallin damage.
  • Predicted impact of adding GSR and 25HC on the amount of crystallin damage in the lens.
  • Created Nanoparticle Customizer for doctors to find a full treatment plan.
  • Generalized our Customizer to allow other iGEM teams who wish to use nanoparticle drug delivery
  • Analyzed sensitivity of prototype, and suggested insights into optimal manufacturing and clinical use of our prototype.
  • Used Experimental data to develop Models 1 and 3.

Outline

Introduction

Why Model?

In the lab, biologists are often unable to test everything experimentally. For example, in our cataracts project, cataract prevention occurs in the long-term, from 20-50 years. Obviously, although short experiments can provide us an idea of what prevention may look like, the power of computational biology allows us to model into the future. As a result, our modeling has been crucial in developing a prototype.

Focus

Most iGEM teams perform modeling on gene expression, which we accomplish in model 5. However, as our construct is not directly placed into the eyes, how our synthesized protein impacts the eyes after it is seperately transported is much more interesting. As a result, we spent the majority of our models on understanding the impacts on the eye.

Guiding Questions

  1. How much GSR do we want inside the lens?
  2. How do we use nanoparticles to control the amount of GSR in the lens?
  3. How do we synthesize GSR, package into NP, and send it into the eye?

Model 1: Crystallin Damage

Abstract

In our experiments, absorbance measurements are meaningless without understanding how severe a cataract that absorbance measurement means. We use literature research to relate LOCS, the physician's scale of cataract severity) to absorbance, which is how we quantified crystallin damage in experiments. We use experimental data to understand how crystallin damage can be quantified by measuring absorbance. With this model, we can calculate how much crystallin damage we have to limit to reduce LOCS to an acceptable level.

Purpose

How much do we need to limit crystallin damage so surgery is not needed?

Measurement of Cataract Severity

There are four 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.
  2. Opacity (%): This is the physical, quantitative property of the LOC scale.
  3. Absorbance at 397.5 nm: This is the experimental method, used by our team in the lab (c.d.).
  4. 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 crystalline is exposed to oxidizing agents. We define 1 crystallin damage unit as the damage done to human crystallin when exposed to 1 M hydrogen peroxide, the main oxidizing agent, for 1 hour.

<

We use the unit of crystallin damage to connect cataract severity with the amount of GSR we add (in Model 2). We want to lower c.d. below so that the resulting cataract is of LOCS 2.5. For the rest of the model, our task is simple: relate each point of the LOCS scale to c.d., in order to connect to Model 2.

LOCS Equivalence to Absorbance: Literature Research

Past studies have done numerous studies on how absorbance measurements can be converted to the LOC scale that physicians end up using. With the results of ________ and ________, we construct the first three columns in Table 2.

Absorbance Equivalence to Crystallin Damage: Experimental Data

We use experimental data from our team’s Cataract Lens Model (link). In each trial, they added H2O2 to crystallin, and measured the resulting absorbance. The data used are shown in Table 1. We can calculate the theoretical c.d., and graph absorbance vs. crystallin damage in Figure 2.

With this relation in Figure 2, we calculate the equivalent crystallin damage to each LOCS rating and its equivalent absorbance.

Error Analysis

It may be surprising that only around 1 M-h is required to induce moderately severe cataracts. Remember that this is done in the absence of antioxidation systems (GSR) and at an extremely high oxidizing concentration of H2O2 (1M of H2O2). In the lens, H2O2 has a much lower concentration, so severe cataracts are induced over months to years.

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 (Assumption 4). 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.

Table 2: Results of Model 1 – Equivalent values for LOCS, Opacity, Absorbance, and Crystallin Damage.
LOCS Opacity (%) Absorbance (@397.5 nm) Crystallin Damage (M-h)
0.0 0.00 0.0000 0.0000
0.5 3.24 0.0143 0.1327
1.0 6.65 0.0299 0.2774
1.5 10.81 0.0497 0.4610
2.0 15.88 0.0751 0.6966
2.5 21.95 0.1076 0.9981
3.0 29.07 0.1492 1.3840
4.0 46.37 0.2706 2.5101
5.0 66.05 0.4691 4.3514

Background

There are four ways to measure cataract severity (how blurred the lens is):

  1. Lens Optical Cataract Scale III (LOCS) - a scale from 0-6 used by physicians.
  2. Opacity (%) - used to calculate the LOC scale
  3. Absorbance at 397.5 nm - measurable in the lab.
  4. Crystallin Damage - used to quantify how much crystallin has been reacted with hydrogen peroxide to create insoluble, damaged crystallin. The following definition of crystallin damage is used:
\[c.d.(t) = \int_{0}^{\infty} [H_2O_2]_t dt\]

In other words, 1 unit of crystallin damage, in M-h, is equal to the damage caused by 1 molar concentration of hydrogen peroxide reacting crystallin in the eyes for 1 hour.

LOCS Scale











Assumptions

  1. Definition of crystallin damage: Crystallin damage is proportional to the concentration of hydrogen peroxide, and the time of exposure. This is a valid assumption, supported by the fact that the reaction between cysteine (molecules on crystallin) and hydrogen peroxide is linear.
  2. We assume that the amount of crystallin is far greater than the amount oxidized. Our product is meant for long-term cataract prevention and minor treatment, and is not suggested for patients with extremely severe cataracts.
  3. 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.
  4. We assume that fish and human lens contain similar crystallin proteins.

Procedure

  1. In the first part, we find how the absorbance measurements in the lab are related to the severity of the cataracts. Through literature data, we can relate LOCS to the opacity of the lens. Then, via physical calculations, we can relate the opacity of the lens to the absorbance of the lens at 400 nm.
  2. Then, we use our team’s experimental data in the cataract model. For each trial, the concentration of H2O2 and the length of exposure are given, so we can calculate the theoretical crystallin damage using the definition above and the assumptions we made. In each trial we also measured the absorbance, so we have a relation between crystallin damage and absorbance.
  3. However, we need to make a minor adjustment, because absorbance is affected by dilution. When the fish lens was isolated, they were placed in Tris buffer and diluted. We calculate the ratio of volumes from diluted volume to the Tris buffer, and multiply each absorbance measurement by this value.

LOCS Scale











Results

Table 1: Results of Model 1 - Equivalent values for LOCS, Opacity, Absorbance, and Crystallin Damage.
LOCS 0.0 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0
Degree None Trace Mild Moderate Severe
Opacity (%) 0 3.24 6.65 10.81 15.88 21.95 29.07 46.37 66.05
Absorbance (a.u.) 0.0000 0.0143 0.0299 0.0497 0.0751 0.1076 0.1492 0.2706 0.4691
Crystallin Damage (c.d.) 0.0000 0.1327 0.2774 0.4610 0.6966 0.9981 1.3840 2.5101 4.3514



Table 2: 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

Discussion

Model Result

The model successfully relates LOCS, opacity of lens, absorbance measurements, and the equivalent crystallin damage of the lens. The purpose of relating LOCS to crystallin damage, is that in Model 2, we will use chemical kinetics to determine how adding GSR to the lens will decrease the amount of crystallin damage. Exactly how much crystallin damage we need to decrease is determined by the desired LOCS. For example, if we want to have a LOCS rating of less than 2.5, then we must lower crystalline damage to only 0.9981 M-h.






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 2, 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.

Error Analysis

It may be surprising that only around 1 M-h is required to induce moderately severe cataracts. Remember that this is done in the absence of antioxidation systems (GSR) and at an extremely high oxidizing concentration of H2O2 (1M of H2O2). In the lens, H2O2 has a much lower concentration, so severe cataracts are induced over months to years.

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 (Assumption 4). Also, to simplify the experiments, the lens were diluted in Tris buffer. Because of this dilution, the actual crystallin damage is much lower, but so is the actual absorbance. We assume that the decrease in crystallin damage and absorbance is the same, so no adjustments need to be made for the relation between crystallin damage and absorbance (Assumption 3). 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.

Conclusion






For surgery to not be needed, the LOCS value has to be below 2.5. This is equivalent to 21.95% in light opacity or 0.1076 abs units. Based on the results of our experiments, this is equivalent to 0.9981 units of crystallin damage, the damage done to crystallin if exposed to 0.9981 M of H2O2 for 1 hr. For future models, this value 0.9981 units of c.d. will be called the crystallin damage threshold for LOCS 2.5.

Model 2: GSR/25HC Chemical Pathway

Abstract

The key question: How much GSR to add? Now that we know how much we need to limit crystallin damage, we use systems of ordinary differential equation to model the GSR Pathway. We calculate the necessary GSR concentration to be maintained over 50 years so that the resulting cataract is below LOCS 2.5./p>

Purpose

How much GSR do we need to maintain in the lens so that the crystallin damage recorded over 50 years is below the threshold for LOCS 2.5?

Chemical Kinetics Model: Differential Equations

By the Law of Mass Action, Michaelis-Menten Enzyme kinetics, Ping-pong mechanism, and the Law of Passive Diffusion, 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

Image

We will vary the input, Initial GSR concentration, from 0 to 100 uM, holding all other variables constant, and numerically solve for the amount of hydrogen peroxide over time. With this graph, we can find the amount of crystallin damage accumulated over 20 to 50 years if different levels of GSR is maintained.

Background

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.

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.

$$\frac{d[GP_{xr}]}{dt}=k_3[GS-GP_x][GSH]-k_1[GP_{xr}][H_2O_2]_{in}$$ $$\frac{d[H_2O_2]}{dt}=k_6[[H_2O_2]_{out}-{H_2O_2]_{in}]-k_1[GP_{xr}][H_2O_2]_{in}$$ $$\frac{d[H_2O_2]}{dt}=k_1([H_2O_2]_{out} - [H_2O_2]_{in})-k_1[GP_{xr}][H_2O_2]_{in}$$ $$\frac{d[GP_{x0}]}{dt}=k_1[GP_{xr}][H_2O_2]_{in}-k_2[GP_{xo}][GSH]$$ $$\frac{d[H_2O]}{dt}=k_1[GP_{xr}][H_2O_2]_{in}+k_2[GP_{xo}][GSH]$$ $$\frac{d[GSH]}{dt}=2k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}-k_2[GP_{xo}][GSH]-k_3[GS-GP_x][GSH]$$ $$\frac{d[GS-GP_x]}{dt}=k_2[GP_{xo}][GSH]-k_3[GS-GP_x][GSH]$$ $$\frac{d[GSSG]}{dt}=k_3[GS-GP_x][GSH]-k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}$$ $$\frac{d[NADPH]}{dt}=-k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}$$ $$\frac{d[GSR]}{dt}=k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}-k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}$$ $$\frac{d[GSR']}{dt}=k_4[GSR]\frac{[NADPH]}{K_{4M}+[NADPH]}-k_5[GSR']\frac{[GSSG]}{K_{5M}+[GSSG]}$$

Part 1 Results

Table 1: Data obtained from XXX relating each value of the LOCS scale, to opacity values.
LOCS 0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0
Degree None Trace Mild Surgery Suggested Moderate Severe Very Severe
Opacity (%) 0.34 4.24 5.80 18.88 23.60 49.14 65.61 90+
Absorbance (a.u.) 0.001 0.019 0.026 0.091 0.117 0.294 0.464 1.3+

We wish to remain below clinically significant levels, so we will reach attempt to lower the LOCS rating of a cataract to below grade 2.5, which means we want to control GSR such that the crystallin damage results in less than 0.108 a.u. at absorbance at 397.5 nm.

Menu 3

Eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo.






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 2, 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.

Conclusion

Conclusion

Model 3: Nanoparticles

Abstract

Abstract

Purpose

Purpose

Background

There are three quantifiers of how severe cataract formation is, two are measurable, one is not.

  1. Absorbance @ 397.5 nm, which is measured with lab equipment.
  2. LOCS scale, subjectively measured by physicians on a scale from 0 - 6.
  3. Crystallin Damage, which we define as the following (for any time $t$)

\[c.d.(t) = \int_{0}^{\infty} [H_2O_2]_t dt\]

In other words, 1 unit of crystallin damage, in M-h, is equal to the damage caused by 1 molar concentration of hydrogen peroxide reacting crystallin in the eyes for 1 hour.

In making this definition, we assume that crystallin damage is directly proportional to the amount of time crystallin is exposed to hydrogen peroxide. Hydrogen peroxide causes damage by forming disulfide bridges within cysteine molecules on crystallin. This changes the structure of crystallin, causing misfolding and cataract damage. Our linear assumption is valid because the rate for this reaction is first order with respect to hydrogen peroxide concentration.

Method

We will relate the three by doing the following:

  1. Relate LOCS scale to opacity via literature research.
  2. Relate opacity to light transmittance via literature research.
  3. Relate light transmittance to absorbance via physical calculations.
  4. Relate absorbance to crystallin damage via experimental data.

Part 1 Results

Table 1: Data obtained from XXX relating each value of the LOCS scale, to opacity values.
LOCS 0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0
Degree None Trace Mild Surgery Suggested Moderate Severe Very Severe
Opacity (%) 0.34 4.24 5.80 18.88 23.60 49.14 65.61 90+
Absorbance (a.u.) 0.001 0.019 0.026 0.091 0.117 0.294 0.464 1.3+

We wish to remain below clinically significant levels, so we will reach attempt to lower the LOCS rating of a cataract to below grade 2.5, which means we want to control GSR such that the crystallin damage results in less than 0.108 a.u. at absorbance at 397.5 nm.

Menu 3

Eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo.

Conclusion

Conclusion

Model 4: Eyedrops

Abstract

Abstract

Purpose

Purpose

Background

There are three quantifiers of how severe cataract formation is, two are measurable, one is not.

  1. Absorbance @ 397.5 nm, which is measured with lab equipment.
  2. LOCS scale, subjectively measured by physicians on a scale from 0 - 6.
  3. Crystallin Damage, which we define as the following (for any time $t$)

\[c.d.(t) = \int_{0}^{\infty} [H_2O_2]_t dt\]

In other words, 1 unit of crystallin damage, in M-h, is equal to the damage caused by 1 molar concentration of hydrogen peroxide reacting crystallin in the eyes for 1 hour.

In making this definition, we assume that crystallin damage is directly proportional to the amount of time crystallin is exposed to hydrogen peroxide. Hydrogen peroxide causes damage by forming disulfide bridges within cysteine molecules on crystallin. This changes the structure of crystallin, causing misfolding and cataract damage. Our linear assumption is valid because the rate for this reaction is first order with respect to hydrogen peroxide concentration.

Method

We will relate the three by doing the following:

  1. Relate LOCS scale to opacity via literature research.
  2. Relate opacity to light transmittance via literature research.
  3. Relate light transmittance to absorbance via physical calculations.
  4. Relate absorbance to crystallin damage via experimental data.

Part 1 Results

Table 1: Data obtained from XXX relating each value of the LOCS scale, to opacity values.
LOCS 0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0
Degree None Trace Mild Surgery Suggested Moderate Severe Very Severe
Opacity (%) 0.34 4.24 5.80 18.88 23.60 49.14 65.61 90+
Absorbance (a.u.) 0.001 0.019 0.026 0.091 0.117 0.294 0.464 1.3+

We wish to remain below clinically significant levels, so we will reach attempt to lower the LOCS rating of a cataract to below grade 2.5, which means we want to control GSR such that the crystallin damage results in less than 0.108 a.u. at absorbance at 397.5 nm.

Menu 3

Eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo.

Conclusion

Conclusion

Conclusion

Yay

Citations












Prevention

GSR Eyedrop

Treatment

25HC Eyedrop

LOCS: 0

Eyedrops