Fusion peptides are increasingly being explored in cancer peptide drug research. It represents a promising area of research in cancer therapy, offering the potential for more targeted, effective, and versatile treatments, due to their ability to enhance the delivery and efficacy of therapeutic agents [1] .
One of the significant challenges in cancer treatment is drug resistance. Fusion peptides can help overcome this by combining different functional domains. For instance, a fusion peptide might include a cell-penetrating peptide (CPP) like TAT or R8, which facilitates the entry of the therapeutic agent into cancer cells, along with a therapeutic domain that induces cell death [2]. By fusing these targeting sequences with therapeutic peptides, researchers can improve the specificity and penetration of the drug into the Tumour Micro-Environment (TME).
In the field of cancer research, predicting the anticancer activity of peptides is a crucial step towards developing effective therapeutic agents. Our modeling work focuses on leveraging computational techniques to forecast the potential efficacy of peptide drugs against various cancer cell lines. By integrating data from peptide sequences, structural properties, and biological assays, we aim to create a robust predictive model that can identify promising candidates for further experimental validation.
In structural modeling, artificial intelligence is used to predict the three-dimensional structures of molecules, their interactions, and their behavior under different conditions. The model employs machine learning algorithms to analyze patterns and correlations within the dataset, enabling us to predict the likelihood of anticancer activity. This approach not only accelerates the drug discovery process but also reduces the reliance on extensive in vitro and in vivo testing, thereby saving valuable time and resources. In our model, a series of de novo design peptide targeting the KRAS signaling pathway, a critical regulator in many cancers, including lung and colorectal cancers, is constructed.
This modeling endeavor aims to facilitate the identification of novel peptide-based therapeutics that can selectively target cancer cells while minimizing the adverse effects on healthy tissues. The ultimate goal is to enhance the precision and efficacy of cancer treatments, paving the way for more personalized and effective therapeutic strategies.
To keep it simple to understand, our modelling work is summarized to have three crucial steps:
Part A: Structural modelling of de novo peptide design: The theoretical background and proposed mechanism of our de novo design peptide drugs KAPI (KRAS and PDEδ Inhibitor)
Part B: Bioinformatic studies Predicting anti-cancer activity of the cell penetrating activity of our list of peptide drug to provide insight for our dry lab
Part C: Exploration of KAPI's property. With wet lab data, KAPI is found to be the most promising ACP. Hence, its properties need further investigation
Part D: Ordinary Differential Equations (ODE) model: With wet lab data, the treatment of the chemotherapy for this in-vitro cancer culture is optimized.
Our objective is to develop a fusion protein that exhibits significant anticancer properties as well as enhanced cell penetration capabilities. In our modeling, dry lab construct model of de novo peptide, predict their potency and provide insight guiding our wet lab investigation. In turn, wet lab verify the prediction and provide data for the dry lab team to construct the ODE model for determining the optimal treatment for this in-vitro cancer cell culture.
Now let’s us guide you to our fantastic journey of discovery through mathematical modeling!
Background: KRAS is unideal to be drugged against; PDEδ is an alternative to alter the KRAS signaling pathway
Our objective: We aimed to generate de novo anticancer peptide sequences that can potentially interact with PDEδ to block the signaling pathway of KRAS, a major pathway involved in non- small cell lung cancer (NSCLC).
Key step 1: RFdiffusion is used to generate 16 backbones of the potential peptide binder to PDEδ
Key step 2: ProteinMPNN is used to design the sequences of those generated backbones (64 sequences for each backbone — so 64*16 = 1024 sequences)
Key step 3: AlphaFold2-multimer is used to screen the 1024 peptide sequences generated from ProteinMPNN.
Key step 4: Using specific metrics, in this case, the lowest PAE from each design (the 16 backbone and the corresponding 64 sequences (one from each)), is deemed the candidate for PDEδ binder.
KRAS is one of the most frequently mutated oncogenes in cancer, accounting for 17~25% of all cancers. In NSCLC specifically, KRAS mutation has a frequency of around 30%[3], and the most common mutation in NSCLC KRAS mutation are G12C (13.43%), which is of common occurrence in current and former smokers. and G12D (29.19%, 13.03% in NSCLC)[4], these oncogenic KRAS mutations develop unique metabolic dependencies on nutrients to support tumor metabolism and cell proliferation.
KRAS alternates between “on” and “off” conformations, differing by the binding with GTP and GDP respectively, under normal conditions. The oncogenic form of KRAS partially inhibits the activity of GTPase, a protein responsible for converting GTP to GDP. This results in a constant active state of KRAS, which tends to prefer binding with GTP over GDP. As a result, GTP-bounded KRAS is able to engage in cytoplasmic signal transduction constantly, most commonly through downstream pathways RAL, RAF-MEK-ERK, and P13K[5], directly resulting in cancer cell survival and abnormal proliferation, and this can be done without a growth factor. Increasing the level of GTP-bound KRAS up to 2-fold, indicating that the identity of the replacing amino acid determines the GTPase activity of oncogenic KRAS[6].
KRAS has been deemed a challenging therapeutic target, even “undruggable”, after drug-targeting efforts over the past four decades[7]. Various studies have previously deemed KRAS an unideal candidate for cancer treatment. This is due to its unusually smooth structure and lack of significant, otherwise shallow drug-binding sites.
After intense research, the spotlight of this project is shifted to another contributor to KRAS signaling: the phosphodiesterase-delta (PDEδ). Signal transduction of KRAS in its active form is highly dependent on its localisation to the cell membrane. Although KRAS protein naturally binds to the cell membrane, this action is usually disrupted by endocytosis, and its distribution is affected by entropy-driven re-equilibration. As a result, KRAS has a tendency to self-aggregate in the cytoplasm.
PDEδ facilitates the proper localization of KRAS to the plasma membrane, which is essential for its activation and subsequent downstream signaling. Moreover, it regulates cyclic AMP (cAMP) levels by hydrolyzing it, thereby influencing critical signaling pathways involved in cell proliferation and survival. Therefore, PDEδ can significantly impact the dynamics of KRAS signaling[8].
PDEδ acts as a solubilisation factor which binds to the farnesyl-tail of KRAS. Its hydrophobic pocket can interact with a farnesylated hydrophobic cysteine residue at the C terminus of KRAS, hence enabling its movement and diffusion in the cytoplasm, and sustaining its spatial distribution on the cell membrane.
Our project uses the preexisting experimentally solved structure of PDEδ (PDB ID: 4JV6) as the starting point of the structure-based design of de novo peptide binder. PDEδ is comprised of 150 amino acids and it has a beta-sandwich immunoglobulin fold, which contains a hydrophobic pocket capable of binding to the farnesyl group [9, 10, 11]
Through literature review, past researchers identified some small molecules that can interact with PDEδ and disrupt the KRAS. It is reported that interfering with the binding of mammalian PDEδ to KRAS by means of small molecules provides a novel opportunity to suppress oncogenic RAS signalling by altering its localization to endomembranes. The important residues are Y149, C56, and R61. Therefore, our project is designated to aims for using AI to design de novo peptides to target the same hotspot residues.
From the above sections, “hotspot” residues on PDEδ are determined, and RFdiffusion is used to generate peptide backbones that could potentially bind to the target residues.
RFdiffusion is an open-source tool designed to generate protein backbones[13]. Its flexibility extends to several advanced applications, including binder design, where it can be used to create proteins that bind specifically to target molecules that fit our project’s requirements.
By simply entering the length of the binder, contigs, hotspots, and the structure of 4JV6, RFdiffusion can generate as much protein backbone as our project needs. Ultimately, our project used RFdiffusion to generate 16 peptide backbones, each 20aa in length, for the following stages for further investigation.
After the peptide backbones are generated in RFDiffusion, ProteinMPNN is used to generate the sequences of the 16 peptides.
ProteinMPNN is a deep learning–based protein sequence design method that performs outstandingly in silico and experimental tests. On native protein backbones, ProteinMPNN recovers 52.4% of the sequence compared with 32.9% for Rosetta[14].
The generated backbones from RFDiffusion are provided to ProteinMPNN to predict their sequences. 64 sequences are generated for each backbone, and 1024 (64*16) sequences are subsequently passed to AlphaFold2-multimer for further screening.
AlphaFold2 is a deep learning model developed by Google Deepmind that allows us to predict a protein’s 3D structure from its amino acid sequence. Using the multimer mode[15], one can possibly predict protein-protein(peptide) interactions
AlphaFold2 was the top-ranked protein structure prediction method by a large margin, producing predictions with high accuracy. While the system still has some limitations, the CASP results suggest AlphaFold2 has immense potential to help us screen the potential protein binder.
In our project, AlphaFold2 has been utilized to predict the most probable sequences bound to PDEδ; for each predicted PDEδ-peptide complex, a Predict Aligned Error (PAE) is given. It represents the overall distance error of each predicted coordinate, i.e. the distance between the predicted coordinate and the actual coordinate.
For each of the 64 sequences predicted from backbones, the one with the lowest PAE score is selected in order to narrow down the number of possible working peptides. The 16 selected sequences are sent to the wet lab for further analysis.
Sequences selected are shown here:
The 3D diagram of the 16 de novo design peptide are shown as follows:
Our objective: The potential anti-cancer peptides are screening with AI database to predict its activity in cancer viability inhibition.
Key step 1: ENNACT and AcPEP is used for anti-cancer activity’s prediction
Key step 2: CellPPD is used for evaluating ACP’s cell-penetrating activity
Key step 3: Rationale behind our fusion peptide
Key step 4: Prediction of anti-cancer and cell-penetrating activity of our fusion peptide
Apart from the de novo peptide design, we are working on another anti-cancer peptide, i.e. ACP19 and 4 peptides identified from Cordyceps militaris.
Through screening methods, potential peptides with strong anti-cancer ability is predicted, allowing us to focus on a reasonable number of peptides in our wet lab. Our approach involves using AI based web tools such as databases and algorithms to study the anticancer peptide (ACP) bioactivity. This helps to facilitate a comprehensive understanding and prediction of the peptides’ effectiveness.
Utilizing a dataset that has been experimentally validated, it is possible to analyze the complete sequences of ACPs to predict their anticancer and cell-penetrating activities through machine learning algorithms. The database ENNACT and AcPEP are employed to assess anticancer potential. Upon inputting the amino acid sequence into the database, a probability score is generated, indicating the likelihood of the sequence being an ACP based on comparisons with existing data. A higher probability score correlates with an increased likelihood of the sequence being classified as an ACP. The results are then analyzed within the database to identify potent peptides that consistently yield high scores in functional prediction.
ENNAACT is a peptide screening database featuring extensive peptide libraries and high- throughput screening capabilities. It employs advanced structure-based virtual screening (SBVS) methods for precise identification of bioactive peptides. ENNAACT’S scanning properties include efficient peptide-protein interaction analysis and rapid dentification of therapeutic candidates[14].
ACPEP is a peptide screening database that identifies anticancer peptides (ACPs) using deep learning models. It scans properties like biological activity against various cancer cells (e.g. breast, colon, lung) through convolutional neural networks and multitasking learning. Key features include its ability to screen properties like amino acid composition, hydrophobicity, and structural motifs. This approach enhances prediction accuracy and aids in rational peptide design for therapeutic purposes[15].
The result is as follows:
The table lists several peptide sequences along with their anticancer activity scores, measured by AcPEP and ENNACT values. Here’s a brief interpretation of the data:
ACP19: This peptide shows the highest anticancer activity with both AcPEP and ENNAACT values at 1, indicating strong potential as an anticancer agent.
C-ori: This peptide has moderate activity with scores of 0.751 (AcPEP) and 0.258 (ENNAACT).
C-rds: The redesigned version of C-ori shows reduced activity compared to its original form, with scores of 0.593 (AcPEP) and 0.196 (ENNAACT).
CTP-rds: This peptide has relatively high score in ENNAACT
These results suggest that the peptide ACP5 (KAPI) has the highest potential for anticancer activity among the listed sequences, particularly due to its high ENNAACT score. Further investigation and optimization of this peptide could be beneficial for developing effective anticancer therapies.
These results highlight the importance of peptide sequence design in enhancing anticancer activity. The scores suggest that these peptides may be promising candidates for further investigation in our anticancer research.
The tool CellPPD is designated for evaluating cell-penetrating activity. A higher probability score correlates with an increased likelihood of the sequence being classified as Cell penetrating peptides (CPPs). The database - Cell penetrating peptides (CPPs) are crucial for delivering various therapeutic cargoes intracellularly. This in-silico SVM-based models predict CPPs with high accuracy of 97.4% , using features like amino acid composition and motifs[16].
Here’s an interpretation of the cell-penetrating activity (CellPPD) data for the peptide sequences:
Generally, all peptide has low ability to penetrate cells. Given the proposed intracellular mechanism of the de novo design peptide, the situation is worrying.
These results suggest that the peptide generally shows poor cell-penetrating activity, Further optimization may be needed to enhance the cell-penetrating capabilities of these peptides.
Cell penetrating peptides have gained much recognition as a versatile transport vehicle for the intracellular delivery of wide range of cargoes (i.e. oligonucelotides, small molecules, proteins, etc.), that otherwise lack bioavailability, thus offering great potential as future therapeutics[16].
In order to improve the cell penetrating activity of our ACPs, a Cell-Penetrating Peptide should be added. The schematic diagram is as follows:
Linker: The flexible linker ‘GGSGGGSG’ in between the amino acids and the Cell-Penetrating Peptide is used. We have taken a reference from the iGem Registry of Standard Biological Parts (Part:BBa_K243005). The linker is designed by Freiburg Bioware from iGEM09_Freiburg_bioware in year 2009[17].
Polyarginine Cell-Penetrating Peptide: Peptides, especially intracellular functional peptides that can play a particular role inside a cell, have attracted attention as promising materials to control cell fate. However, hydrophilic materials like peptides are difficult for cells to internalize[18]. Octaarginine (R8) as a Polyarginine Cell-Penetrating Peptide (CPP) is reported to enhance peptide’s ability to cross the cell membrane and deliver cargo[19]. Hence, ‘RRRRRRRR’ is used as our Cell-Penetrating Peptide sequence for penetrating cell membrane and enter the cell,.
The backbone of the fusion peptides is finalized. It is designed to have promising ability to target intra-cellular PDEδ and enter cancer cells effectively.
The fusion peptides are finally screened by the AI database to check on their anti-cancer and cell penetrating activity.
ACP5 (KAPI) shows the highest anticancer activity (AcPEP: 1) and also has a relatively high cell-penetrating activity (CellPPD: 0.74).
CTP-rds from Cordyceps militaris also shows high anticancer activity (AcPEP: 0.988) and decent cell-penetrating activity (CellPPD: 0.56).
Generally, the de novo designed peptides tend to have higher anticancer activity compared to the peptides from Cordyceps militaris.
KAPI, i.e. ACP5, seem very promising due to their high anticancer and cell-penetrating activities. This suggests that they could be effective candidates for further development and testing in cancer therapy.
In conclusion, by incorporating cell-penetrating peptides into the ACPs, the resulting fusion peptide demonstrates an enhanced ability to infiltrate cancer cells. As a result, the overall efficacy in anticancer activity should be significantly increased.
In the previous section, the anti-cancer ability (AcPEP and ENNACT) and cell penetrating ability (CellPPD) is predicted with assistance from AI database. Combining the wet lab study of all ACP, KAPI is found to be the most promising ACP. Hence, its property is further investigated, and the overview is as follows (Figure 8)
Finding from the other database is summarized:
The pH of human blood is normally slightly alkaline, with a range of 7.35 to 7.4512. This narrow range is crucial for maintaining proper physiological functions and is tightly regulated by the body’s buffering systems, including the lungs and kidneys. Taking pH 7.4 as a reference [20], the peptide solubility is estimated by CamSolpH. CamSolpH yields a solubility profile (one score per residue in the protein sequence) where regions with scores larger than 1 denote highly soluble regions, while scores smaller than -1 poorly soluble ones. An overall solubility score will be assigned to the whole sequence. This score can be used to rank with high accuracy different protein variants according to their solubility [21]. KAPI’s intrinsic solubility score at pH7 is 3.073575.
The plot of the solubility profile is shown:
ProtParam is a tool for calculating various physical and chemical parameters of a protein sequence, such as molecular weight, theoretical pI, and amino acid composition [22]. The calculated value of KAPI is shown:
Theoretical pI: 12.95
Estimated half-life: 30 hours (Taking mammalian reticulocytes, in vitro as an reference)
The instability index (II) is computed to be 103.61. This classifies the protein as unstable.
Protein thermal stability is an important parameter for understanding protein function, designing protein-based therapeutics, and improving the stability of enzymes used in industrial processes [23]. DeepSTABp web interface utilize the deep learning model to predict the melting point of a protein. The calculated value of KAPI is 57.02069068653388°C
AllerTOP v. 2.0 is a bioinformatics tool for allergenicity prediction. Amino acid hydrophobicity, molecular size, helix-forming propensity, relative abundance of amino acids etc. is studied. The proteins are classified based on training set containing 2427 known allergens from different species and 2427 non-allergens [24] . KAPI is defined as non-allergen by the database.
Among the finding from the database, the result from ProtParam is concerning. Given the findings, the instability of the peptide suggests that it may degrade quickly under physiological conditions, potentially limiting its therapeutic efficacy. This data inspires us to develop an Ordinary Differential Equation (ODE) model to better understand and predict the peptide’s behavior over time. By incorporating factors such as degradation rates, cancer growth rate, the ODE model will help us optimize the in-vitro chemotherapy treatment, facing challenge from peptide’s stability and efficacy for potential therapeutic applications.
The development of peptide therapeutics is challenging due to their low stabilities and short half-life. However, peptides typically have exquisite potency, selectivity, and low toxicity, making them particularly attractive for disease targets [25]. Hence, concentration and duration of KAPI treatment as chemotherapy require optimization by Mathematical modelling. In this section, we aim to mimic the optimization of chemotherapy using an in-vitro culture of A549 Lung Adenocarcinoma.
To model the impact of anticancer peptides KAPI on the growth of cancer cell A549, an Ordinary Differential Equation (ODE) model is used. It can illustrate how the state of a system changes over time as a function of its current state and some parameters. It allows for predicting cancer population dynamics over time in response to varying concentrations of anticancer substances. By employing the ODE model and analyzing the obtained results, the necessary quantity of ACPs to be incorporated to hinder cancer cell growth effectively is approximated. In addition, experimental data collected offers valuable insights into the intricate interactions between KAPI and cancer cell growth. Consequently, these insights can guide the development of the in-vitro cancer cell model.
The flow of the ODE page can be summarised as follows:
The construction of the ODE model is based on several assumptions:
1. The interaction of anticancer peptides and cancer cell is regarded as a first-order reaction, this indicates that the rate of cell death is directly proportional to the concentrations of anticancer peptides (KAPI).
2. After KAPI binds to PDEδ, the peptide drug is consumed, and it is assumed the action is not reversible. This irreversible binding leads to the inhibition of PDEδ, which plays a crucial role in the KRAS signaling pathway. By inhibiting PDEδ, KAPI effectively disrupts the localization and function of KRAS, thereby impeding the downstream signaling that promotes cancer cell proliferation and survival. This targeted approach ensures that the therapeutic effects are sustained, as the bound KAPI-PDEδ complex is no longer available to participate in the signaling pathway.
3. In this model, it is assumed that the mixture within the system is homogeneous. That is, the substance being mixed is uniformly distributed throughout the entire volume of the mixture at any given time, where the concentration of the substance is at a uniform distribution within the mixture and the mixing process is assumed to be instantaneous. To simplify things, the volume of the mixture remains constant over time.
4. For simplification, it is assumed that the cell growth follows the Gompertz growth curve. This assumption is based on the following key points. To commence with, the Gompertz model describes a sigmoidal growth curve, which is characterized by an initial exponential growth phase, followed by a deceleration phase, and finally reaching a plateau as the carrying capacity is approached [26]. At the beginning of the growth process, cells proliferate rapidly. This phase is marked by a high growth rate, which gradually decreases over time. As the cell population increases, the growth rate slows down. This deceleration is due to factors such as nutrient depletion and waste product accumulation, which is similar to the culture environment in a 96-well plate. The model assumes that there is a maximum carrying capacity that the environment can support. Lastly, it is generally accepted that the Gompertz model is one of the promising models to predict the growth of tumor in cancer research.
5. It is assumed that both the temperature and pressure within the system remain constant over time. The system is assumed to be in both thermal and mechanical equilibrium in which any changes in temperature or pressure are immediately balanced by heat exchange with the surroundings and pressure adjustments, maintaining constant conditions. The assumption is supported by the environment of the cancer cell culture, which is inside a CO2 incubator with 37oC.
Penultimately, it is assumed that once the anticancer peptide diffuses into the cancer cell, it is subject to degradation by intracellular proteases. The rate of degradation is assumed to follow first-order kinetics, represented by the degradation constant (k_2). The degradation process is uniform throughout the intracellular environment. The peptide’s stability is not influenced by the presence of other intracellular molecules or organelles, except for the proteases responsible for its degradation. The rate of degradation in the culture medium is not considered in the study.
7. Lastly, it is assumed that the anticancer peptide specifically interacts with the PDEδ. This interaction is highly selective, meaning the peptide does not react with other cellular components or pathways. The interaction between the peptide and PDEδ does not alter the degradation rate of the peptide. The degradation and interaction processes are independent of each other.
Based on these assumptions, the following equations are constructed:
The variables involved in the process are as follows:
The equations above correspond the following rate laws:
To estimate the rate constant of the ODE model, four groups of parameters are measured:
A. The rate of diffusion of KAPI into and out of the cancer cells (rate constant k_1and k_(-1))
B. The rate of degradation of KAPI inside cancer cells (rate constant k_2)
C. The cell death rate constant induced by KAPI (rate constant k_3)
D. The proliferation of cancer cells (Gompertz Growth Model)
The k_1and k_(-1) are estimated together with k_3 by the time-lapse viability assay (will be discussed in part C). The estimated k_1 and k_-1 are 1.1922787 h-1 and 0.19613276 h_-1 respectively.
The estimated half-life of KAPI from ProtParam is used to estimate the degradation rate of k_2. ProtParam is a tool that allows the computation of various physical and chemical parameters for a given protein stored in UniProtKB or for a user entered protein sequence [22]. One of them is the estimated intracellular half-life of the input sequence in mammalian reticulocytes. Taking this data as a reference, the half-life of KAPI sequence is estimated to be 30 hours. To determine the rate constant of degradation from an estimated half-life, the data is inputted into the formula for a first-order reaction:
where:
- k_2 is the rate constant,
- ln(2) is the natural logarithm of 2 (approximately 0.693),
- t_(1/2) is the half-life.
Given the half-life (t_(1/2)) is 30 hours, you can calculate the rate constant (k ) as follows:
So, the rate constant of degradation is approximately 0.023196h^(-1).
The rate constant k_3 is estimated by a time-lapse viability assay. A time-lapse viability was conducted to calculate the change in number of cells alive after adding KAPI. The procedures are as follows:
1. 5000 cells with 100μL culture medium were seeded into each well of a 96-well plate.
2. Old medium is removed.
3. 100μL culture medium with 0μM, 25μM and 50μM KAPI was added two days after cell seeding.
4. The cell viability is calculated at a regular time interval after the start of cell seeding (i.e. 0, 12, 24, 36, 48h after initial cell seeding)
5. Data collected is fitted into the ODE model for estimation of rate constant k_1,k_(-1) and k_3.
Hence, the cell death's rate constant k3 is estimated to be 0.33779115 h-1
The Gompertz Model is a mathematical model used to describe population dynamics, particularly in biological contexts [27]. It characterizes growth as a decelerating process, where the rate of growth decreases exponentially over time. The rate constant for Gompertz growth model is estimated from the wet lab data, calculating the change in the number of cells in a 96 well plate after cell seeding. The procedures are as follows:
1. 5000 cells with 100μl culture medium was seeded into each well of a 96-well plate.
2. Cells are allowed to grow for two days.
3. Old medium is removed.
4. Freash medium with 0μM, 25μM and 50μM KAPI is added to each well.
5. The number of cells is calculated at a regular time interval after the start of drug treatment (i.e. 0, 12, 24, 48, 72 and 96h after initial cell seeding) by trypan blue exclusion.
6. Data collected is fitted into the ODE model for estimation of rate constant for the growth model
Let X be the number of cancer cells, with growth constants α and β. The Gompertz model is given by
differentiating with respect to t gives
Change in population of cells over time is fitted into the computational code as follows:
import numpy as np
from scipy.optimize import curve_fit
## Curve fitting
import matplotlib.pyplot as plt
# Gompertz model function
def gompertz_model(t, A, B, X_max):
return X_max * np.exp(-np.exp((A - B * t)))
t = np.array([0, 12, 24, 48, 72, 96])
y_data = np.array([3.979166667, 9.100, 10.675, 13.83666667, 32.66666667, 35.000])
# Fit the Gompertz model to the data
popt, pcov = curve_fit(gompertz_model, t, y_data, bounds=(10e-26, [np.inf, np.inf, np.inf]), x_scale=[1, 1, 10])
# Extract the fitted parameters
A_fit, B_fit, X_max_fit = popt
# Generate the fitted curve
y_fit = gompertz_model(np.linspace(-10, 200, 500), A_fit, B_fit, X_max_fit)
# Plot the data and the fitted curve
plt.scatter(t, y_data, label='Data')
plt.plot(np.linspace(-10, 200, 500), y_fit, label='Fit')
plt.xlabel('Time')
plt.ylabel('Population')
plt.legend()
plt.show()
After all, the growth of cancer cells is modeled as follows:
The experimental data was fitted and the estimated α and β is 9.96207288×10^(-1) h^(-1) and 1.59791807×10^(-2) h^(-1)respectively.
The parameters obtained previously are fitted into an ODE model. The source code is as follows:
from scipy.optimize import minimize, rosen, rosen_der
import numpy as np
from scipy.integrate import odeint
from scipy.optimize import curve_fit
from scipy.integrate import odeint
t = np.linspace(0, 504, 100001)
y0 = [5.8,0,0,5,0]
k1 , k_neg1, k3 = np.array([1.1922787 , 0.19613276, 0.33779115])
#define rate equations
def rate_eq(y, t, k1,k_neg1, k3):
D_5e, D_5i, XD_5, cells, dead_cells= y
k2 = 0.02310490601
A = 9.96207288e-01
B = 1.59791807e-02
dydt = [-k1*D_5e+k_neg1*D_5i,k1*D_5e-k_neg1*D_5i-k2*D_5i-k3*cells*D_5i,k2*XD_5, -k3*cells*D_5i+cells*B*np.exp(A-B*t),k3*cells*D_5i]
return dydt
sol = odeint(rate_eq, y0, t, args=(k1, k_neg1 ,k3))
import matplotlib.pyplot as plt
plt.plot(t, sol[:, 0]*10, 'b', label='[Extracellular KAPI / Ke (Unit:μM)]')
plt.plot(t, sol[:, 1]*10, 'g', label='[Intracellular KAPI / Ki (Unit:μM)]')
plt.plot(t, sol[:, 3]*10, 'r', label='[Number of cancer cells (Unit: 100cells/well)]')
plt.legend(loc='best')
plt.xlabel('Time / h')
plt.ylabel('Ke / Ki / Number of cancer cells')
# plt.yticks(np.arange(0, 700, 700/350))
plt.show()
With the estimated rate constant, the following graphs are constructed by a python program to demonstrate the change in the concentration of KAPI and cancer cells:
Initially, extracellular KAPI diffuses into the cancer cell. When 35µM of KAPI is applied, the concentration of KAPI outside the cancer cell falls drastically after time zero, while the concentration of KAPI inside the cancer cell increases, indicating the diffusion of KAPI into the cancer cell. However, after a duration of less than 5 hours, the concentrations of KAPI inside the cancer cell begin to decline.
In a starting population of 5,000 cancer cells, a sharp decline is observed over time following the administration of KAPI, which demonstrates its capability to inhibit the proliferation of cancer cells. However, the inhibition of cancer growth slowly declines as the concentration of extracellular and intracellular KAPI drop quickly. After 5 hours, a turning point is reached, marked by an increase in the cancer cell population. This phenomenon reflects the degradation of KAPI, resulting in a loss of its inhibitory function. Consequently, cancer cells can proliferate rapidly, leading to a resurgence in their population.
This scenario highlights the inadequacy of the current quantity of ACPs in effectively suppressing cancer cell proliferation, which occurs at a rapid pace. Therefore, it is imperative to tailor the treatment by adjusting the relevant variables.
The focus of our model is to utilize KAPI for inhibition of cancer cell growth, with the challenge from its short half-life inside cells. To determine the appropriate treatment using the ODE model developed, several criteria are considered.
Firstly, the number of KAPI used should be sufficient to inhibit cancer cell viability to an acceptable level after one round of chemotherapy, ensuring an acceptable approach for cancer treatment. Meanwhile, the number of KAPI should be minimized to reduce the development of chemotherapy resistance. Since the development of chemotherapy resistance towards conventional therapy is one of the important reasons for chemotherapy failure in cancer[28], studies have shown that discontinuous dosing and modifying drug concentrations can combat drug resistance and improve patient survival [29].
The second consideration is that the concentration of KAPI should be sufficient to effectively suppress the proliferation of cancer cells throughout a treatment cycle, which typically occurs once every three weeks, referred to as a 21-day regimen [30]. Additionally, based on the recommendation of Dr. Lam Ka On during an interview, it is suggested that the objective for reducing cancer cell viability within a treatment cycle should be a minimum of 90%. The interview inspires us to set target for this in-vitro chemotherapy ODE model in a treatment cycle.
In short, the minimum amount of KAPI needed to inhibit the growth of cancer cells in the in-vitro model so that they are kept at an acceptable standard within 504 hours should be found. Then the following graphs are constructed by adjusting the initial value of KAPI to test out different situations for further interpretation.
In Figure 13, a partial response is illustrated. A partial response is characterized by a minimum 30% reduction in the total diameters of target lesions, indicating our objective to inhibit 30% of the cancer in this scenario [31,32] . A 24µM of KAPI is utilized in the illustration. Following a brief decline, the population of cancer cells exhibits a continuous increase. Concurrently, the depletion of KAPI occurs rapidly. A partial response suggests that the treatment is effective in reducing the tumor burden, but additional treatment may be necessary to achieve further reduction or control of the cancer cell.
In Figure 14, the time scale is extended to 24 hours. It is shown that cancer cells continue to grow, which reflects that the partial response is not effective, until a frequent drug treatment is implemented.
In Figure 15, a significantly elevated quantity of KAPI (100µM) is administered, aiming to achieve complete response [32]. The count of cancer cells diminishes swiftly within the first hour, coinciding with a substantial reduction in the concentration of extracellular KAPI. Following this initial decline, the level of extracellular KAPI decreases at a slower rate.
In Figure 16, it is shown that extracellular KAPI ultimately being depleted by the 200th hour. Regarding intracellular KAPI, it initially rises before experiencing a decline. KAPI stay in the medium and cancer cells for a longer time and ensure no recovery of cancer cells afterwards. Hence, complete response is achieved.
Although excess KAPI is effective in eradicating all cancer cells, it is not considered an optimal dosage because of the following reasons:
1. The concentration of drugs used is extremely high, it is likely to induce chemotoxicity, potentially increase the harmful effect of chemoradiotherapy.
2. Technical problem may arise due to the production of drugs with a high concentration.
3. It is not financially logical for the extreme overkill.
Hence, the optimal treatment should be achieved. In Figure 17, based on the suggestion of Dr. Lam, the starting concentration of KAPI in this scenario is 59μM. It is found that this concentration has been determined to be adequate for achieving a 90% inhibition of the cancer cell population, while maintaining a cancer cell concentration not exceeding 10% of the original number of cancer cells over a period of 252 hours (1.5 weeks). This finding suggests that the specified amount of KAPI is appropriate for a treatment regimen and represents the minimum dosage required to effectively inhibit the proliferation of cancer cells.
- Limitation: To commence with, the time-lapse viability assay is not long enough to support a strong predictive power over a long duration. Also, despite the success in constructing 3D spheroid model, it was not used in the time-lapse viability assay due to financial and time constraints. When tackling with the ODE model, it is found that cancer cells grow so quickly which has a limited representation of the situation in-vivo. In addition, the effect of KAPI is tested alone but not combined with cisplatin.
- Future plan: At least three successive cycles for 21-day regimen should be conducted for the time-lapse viability assay in a 3D spheroid culture so that the model can be more reliable in terms of rate of cancer cell growth and chemotherapy duration. Also, the effect of the combined therapy can be studied. Thus, the limitation of being unable to simulate the tumour microenvironment (TME) for a more accurate investigation can be improved with the application of 3D models for the revised ODE model.
- Limitation: Due to the limited time and apparatus, some rate constants can only either be estimated indirectly (e.g. the rate constant of diffusio) or ignored (e.g. association and dissociation rate constant of KAPI to PDEδ, rate of excretion) in our study.
- Future plan: Taking rate constant of diffusion as an example, it can be determined by fluorescence Microscopy [33]. Peptide labelled by fluorescent dye can be incubated with cancer cells. By capturing images at different time points by Fluorescence microscopy, the fluorescence intensity inside the cells over time can be determined for the calculation of the rate of diffusion. Also, the association and dissociation rate constant of KAPI to PDEδ can be found by Surface Plasmon Resonance (SPR): SPR measures the binding interactions in real-time by detecting changes in the refractive index near the sensor surface where the peptide is immobilized. The association rate constant and dissociation rate constant can be derived from the sensorgram data13.
In our study, an ODE model is developed for simulating an in-vitro chemotherapy treatment. In Figure 18, an identical optimal concentration of KAPI (59μM) is administered, while the objective is to evaluate its efficacy under different initial conditions, specifically with a starting cell count of 5,500 cells at 59μΜ. The results indicate that cancer cells exhibit rapid proliferation following a transient decrease. This observation suggests that a uniform dosage of KAPI may not yield consistent outcomes across all patients, given the variability in tumor size, medical history, sex and age, etc.Consequently, the treatment cycle may not adhere to the standard duration, and the 59μM concentration may not be universally applicable. Ultimately, this in-vitro model can be a valuable tool to guide efforts to enhance personalized medicine approach [34] for an elevated efficiency in chemotherapy treatment.
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