The document presents a compartment-based model to describe the degradation kinetics of a peptide substrate reporter (VI-B) in five cell cultures. The model uses a system of differential equations to track the concentration of the reporter and its fragments over time. Parameters of the model are fitted to time-series data from the cultures. Results show variation in degradation rates between cultures and fragments, revealing targets of peptidases. The model has potential utility in peptide substrate reporter design.

1. COMPARTMENT-BASED MODEL FOR VI-B
PEPTIDE DEGRADATION
Nhat Pham1
, Brooks K. Emerick1
, Michelle L. Kovarik2
, Allison J. Tierney2
, Kunwei Yang2
1
Department of Mathematics,
2
Department of Chemistry, Trinity College
Abstract
A peptide is a chemical compound made up of amino acid bonds
which can be broken down by enzymes in the body, converting the
peptide into smaller chains. Knowing the kinetics of peptide degra-
dation can be useful in designing peptides that resist degradation. In
this research project, we are interested in finding the constant rates
of degradation of a peptide substrate reporter for protein kinase B
(VI-B) in five different cell cultures (Dictyostelium, HeLa, LNCaP,
Yeast, and E.Coli) using data collected by Prof. Kovarik, Allie Tier-
ney, and Kunwei Yang of the Chemistry Department. Based on
the physical process of the peptide, we see that compartment-based model can be used to track
the evolution of peptide strand as time evolves. We fit the solution functions of the model to the
given time-series data sequentially using tools of least-squares regression, and determine the best
fit parameters to be parameter values that yield the minimum residual errors among all trials.
Modeling results show variations in rate of degradation among reactions and organisms, reveal
which amino acid residues and fragments are targeted by peptidases and consequently the method
demonstrate potential utility in peptide substrate reporter design.
Model Formulation
VI-B are readily degraded by peptidases to break down into smaller fragments whose concentra-
tions are measured. In this work, we use compartment-based model to describe the concentration
over time. The model includes several assumptions:
(1) the initial reporter is the only source of fragments
(2) the parent peptide or any small fragments do not leave the system, but are simply converted
to smaller fragments
(3) any larger peptide could be cleaved to form any smaller peptide, but only fragments re-
taining 6-FAM label will be detected
(4) the system follows first-order kinetics and all rate constants are non-negative
Fig.1 Model of reporter (B) metabolism into shorter fragments
from eukaryotic cells with rate constants ki
Fig.2 Reporter concentration as a function of time
The model can be translated into a series of differential equations:
dB
dt
= −(k8 + k7 + k6 + k5)B = −dpB (1)
dB8
dt
= −(k87 + k86 + k85)B8 + k8R = −d8B8 + k8B (2)
dB7
dt
= −(k76 + k75)B7 + k87B8 + k7B = −d7B7 + k87B8 + k7B (3)
dB6
dt
= −k65B6 + k76B7 + k86B8 + k6B (4)
dB5
dt
= k65B6 + k75B7 + k85B8 + k5B, (5)
In E.Coli lysates, only segments VI-B7 through VI-B4 are observed. As a result, a slightly different
compartment-based model was required to describe this system. This model is mathematically
equivalent to the model above, and the same assumptions were applied.
Data Fitting Methods
To find the best fit parameters for each segment, we first solved Equations (1)-(5) using basic
techniques for solving differential equations, then match the explicit solutions to the given time-
series data in the least-squares sense. We first found the best fit parameter for the parent peptide,
B(t), then found best fit parameters for B8, B7, B6, B5 sequentially .
• Fitting data for the reporter (B):
- Find explicit solution to equation (1):
B(t, dp) = e−dpt
- Linearize the function by taking the natural
logarithm ln(B) = −dpt and match it to the
logarithm of the given data.
- Minimize the residual error using the least-
squares regression line and find the best fit for
parameter dp
0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min)
ConcentrationProportion
Data-Fitting (VI-B,LNCaP)
• Fitting data for B8:
- Substitute best fit value for dp to Eq. (2) and solve for B8:
B8(t) = k8
dp−k8
(e−dpt − e−dpt)
- Fit function to the data using non-linear least-squares approach:
+ Set lowerbound and upperbound for parameters: 0 ≤ d8 ≤ 1 and 0 ≤ k8 ≤ dp
+ Run 10,000 simulations in Matlab lsqnonlin function with randomized initial conditions be-
tween [0,1] and report the parameter value that yields the minimum residual error to be best
fit parameter, also considering the most popular value among trials.
0.0657 0.0657 0.0658 0.0658 0.0659 0.0659 0.066
0
1000
2000
3000
4000
Distribution for Parameter d8 (LNCaP)
d8 fit
Frequency
10000 Trials
Min Res = 3.2052e-05
Best d8 = 0.065783
−1.1824 −1.1822 −1.182 −1.1818 −1.1816 −1.1814 −1.1812 −1.181 −1.1808 −1.1806
0
1000
2000
3000
4000
Log Distribution for Parameter d8 (LNCaP)
log(d8) fit
Frequency
10000 Trials
Min Res = 3.2052e-05
Best log(d8) = -1.1819
5.288 5.29 5.292 5.294 5.296 5.298 5.3
x 10
−3
0
1000
2000
3000
4000
Distribution for Parameter k8 (LNCaP)
k8 fit
Frequency
10000 Trials
Min Res = 3.2052e-05
Best k8 = 0.005291
−2.2767 −2.2766 −2.2765 −2.2764 −2.2763 −2.2762 −2.2761 −2.276 −2.2759 −2.2758 −2.2757
0
1000
2000
3000
4000
Log Distribution for Parameter k8 (LNCaP)
log(k8) fit
Frequency
10000 Trials
Min Res = 3.2052e-05
Best log(k8) = -2.2765
• Fitting data for other fragments: Applying similar method to find other best-fit pa-
rameters. Note that B5(t) is completely determined by the previous four solutions.
0 50 100 150 200 250 300 350
0
0.1
0.2
0.3
0.4
0.5
0.6
time (min)
ConcentrationProportion
Data-Fitting (Dictyostelium)
VI−B5
VI−B6
VI−B7
VI−B8
0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min)
ConcentrationProportion
Data-Fitting (LNCaP)
VI−B5
VI−B6
VI−B7
VI−B8
0 10 20 30 40 50 60
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
time (min)
ConcentrationProportion
Data-Fitting (HeLa)
VI−B5
VI−B6
VI−B7
VI−B8
0 50 100 150 200 250 300 350
−0.05
0
0.05
0.1
0.15
0.2
0.25
time (min)
ConcentrationProportion
Data-Fitting (Yeast)
VI−B5
VI−B6
VI−B7
VI−B8
Best-fit Parameters and Errors
Table 1: Best fit parameter values with minimum residual and highest frequency
Parameter
Dictyostelium HeLa LNCaP Yeast E-Coli
Min Res Max Freq Min Res Max Freq Min Res Max Freq Min Res Max Freq Min Res** Max Freq
dp 7.3783e-03 - 7.3096e-03 - 4.4455e-02 - 2.3341e-03 - 2.1500e-02 -
d8 4.2125e-03 4.36e-03 4.4931e-02 4.49e-02 6.5783e-02 6.58e-02 1.7033e-02 1.71e-02 4.3809e-04 4.38e-04
k8 2.2987e-04 2.33e-04 2.8461e-03 2.85e-03 5.2909e-03 5.29e-03 1.5109e-03 1.51e-03 1.2944e-02 1.29e-02
d7 3.9124e-04 3.91e-04 3.0218e-02 3.46e-02 1.0075e-01 1.11e-01 9.4708e-03 9.47e-03 1.8829e-02 1.88e-02
k87 1.2876e-13 2.22e-14 2.0177e-02 2.28e-02 5.4015e-02 4.40e-02 1.7033e-02 1.70e-02 3.3537e-06 3.37e-06
k7 5.4568e-03 5.46e-03 2.3930e-04 2.06e-04 4.7989e-04 6.48e-04 1.7692e-04 1.78e-04 8.5558e-03 8.56e-03
d6 3.2309e-12 6.48e-07 1.4341e-02 1.61e-02 1.0000e-00 1.00e-00 1.5530e-09 1.73e-07 4.6765e-13 9.00e-09
k86 4.2125e-03 4.21e-03 4.3660e-08 2.24e-14 1.3304e-11 2.22e-14 9.7969e-15 9.80e-15 4.3474e-04 4.35e-04
k76 1.2180e-11 2.25e-14 2.8988e-02 3.01e-02 2.3307e-14 2.60e-05 5.0344e-03 5.03e-03 2.2205e-14 2.22e-14
k6 1.5388e-03 1.54e-03 1.9253e-04 2.56e-04 1.8884e-02 1.89e-02 2.2303e-14 2.22e-14 8.4237e-08 8.42e-08
k85 2.2204e-14 - 2.4754e-02 - 1.1768e-02 - 2.2232e-14 - 2.2205e-14 -
k75 3.9124e-04 - 1.2300e-03 - 1.0075e-01 - 4.4364e-03 - 1.8829e-02 -
k65 3.2309e-12 - 1.4341e-02 - 1.0000e-00 - 1.5530e-09 - 4.6765e-13 -
k5 1.5282e-04 - 4.0318e-03 - 1.9801e-02 - 6.4625e-04 - 2.2206e-14 -
**EColi’s parameter sequence:dp,d7, k7, d6,k76,k6,d5,k75,k65,k5,k74,k64,k54
* Min Res and Max Feq values with more than two orders of magnitude difference are in red
Table 2: Minimum residuals for each peptide and species
Segment Dictyostelium HeLa LNCaP Yeast E-Coli
VI-B 9.3214e-02 2.1014e-02 7.9433e-02 1.1629e-01 1.4732e-01
VI-B8 2.7728e-04 9.7549e-05 3.2052e-05 3.0242e-04 -
VI-B7 2.3038e-03 6.5266e-06 1.1345e-05 6.8960e-04 2.3609e-02
VI-B6 4.2415e-03 2.2384e-04 4.6258e-03 1.6917e-02 1.2882e-03
VI-B5 4.2415e-03 2.2384e-04 4.6258e-03 1.6917e-02 1.9770e-01
VI-B4 - - - - 1.9770e-01
General Model
The simplified general model for the system of ODEs written in matrix form:
dy
dt
=






−α1 0 0 0 0
α1/4 −α2 0 0 0
α1/4 α2/3 −α3 0 0
α1/4 α2/3 α3/2 −α4 0
α1/4 α2/3 α3/2 α4 0






y, y(0) =






1
0
0
0
0






.
General solution: y(t) = C1v1e−α1t + C2v2e−α2t + C3v3e−α3t + C4v4e−α4t + C5v5 , where
v1 =








4(α1−α2)
4α2−3α1
3(α1−α3)
3α3−2α1
2(α1−α4)
2α4−α1
α1
4α2−3α1
−3(α1−α3)
3α3−2α1
2(α1−α4)
2α4−α1
α1
3α3−2α1
2(α1−α4)
2α4−α1
−α1
2α4−α1
1








, v2 =







0
−3(α2−α3)
3α3−2α2
2(α2−α4)
2α4−α2
α2
3α3−2α2
2(α2−α4)
2α4−α2
−α2
2α4−α2
1







, v3 =






0
0
2(α3−α4)
2α4−α3
−α3
2α4−α3
1






, v4 =




0
0
0
−1
1



 , v5 =




0
0
0
0
1




• Future work: We would like to generalize the compartment-based model to any number of
peptide fragments - to develop the general solution to this ODE system in n-dimension.
Conclusions
• Rate constants for the degradation reactions varied widely between reactions and organisms.
Overall, the degradation resistance of VI-B suggests that it is sufficiently stable for application
across eukaryotic cells (with the exception of the rapid degradation in LNCaP lysates).
• Given only ten data points for each peptide strand, we found it more economical to fit the data
in a sequential order rather than all at the same time and the sequential method described
above yielded smaller residual norms.
• These results demonstrate the potential utility of compartment-based models for optimizing
substrate reporters for degradation resistance.
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