A minimal mathematical model for the initial molecular interactions of death receptor signalling

  • Received: 01 August 2011 Accepted: 29 June 2018 Published: 01 July 2012
  • MSC : Primary: 92C37.

  • Tumor necrosis factor (TNF) is the name giving member of a large cytokine family mirrored by a respective cell membrane receptor super family. TNF itself is a strong proinflammatory regulator of the innate immune system, but has been also recognized as a major factor in progression of autoimmune diseases. A subgroup of the TNF ligand family, including TNF, signals via so-called death receptors, capable to induce a major form of programmed cell death, called apoptosis. Typical for most members of the whole family, death ligands form homotrimeric proteins, capable to bind up to three of their respective receptor molecules. But also unligated receptors occur on the cell surface as homomultimers due to a homophilic interaction domain. Based on these two interaction motivs (ligand/receptor and receptor/receptor) formation of large ligand/receptor clusters can be postulated which have been also observed experimentally. We use here a mass action kinetics approach to establish an ordinary differential equations model describing the dynamics of primary ligand/receptor complex formation as a basis for further clustering on the cell membrane. Based on available experimental data we develop our model in a way that not only ligand/receptor, but also homophilic receptor interaction is encompassed. The model allows formation of two distict primary ligand/receptor complexes in a ligand concentration dependent manner. At extremely high ligand concentrations the system is dominated by ligated receptor homodimers.

    Citation: Christian Winkel, Simon Neumann, Christina Surulescu, Peter Scheurich. A minimal mathematical model for the initial molecular interactions of death receptor signalling[J]. Mathematical Biosciences and Engineering, 2012, 9(3): 663-683. doi: 10.3934/mbe.2012.9.663

    Related Papers:

    [1] Wenxue Huang, Yuanyi Pan . On Balancing between Optimal and Proportional categorical predictions. Big Data and Information Analytics, 2016, 1(1): 129-137. doi: 10.3934/bdia.2016.1.129
    [2] Dongyang Yang, Wei Xu . Statistical modeling on human microbiome sequencing data. Big Data and Information Analytics, 2019, 4(1): 1-12. doi: 10.3934/bdia.2019001
    [3] Wenxue Huang, Xiaofeng Li, Yuanyi Pan . Increase statistical reliability without losing predictive power by merging classes and adding variables. Big Data and Information Analytics, 2016, 1(4): 341-348. doi: 10.3934/bdia.2016014
    [4] Jianguo Dai, Wenxue Huang, Yuanyi Pan . A category-based probabilistic approach to feature selection. Big Data and Information Analytics, 2018, 3(1): 14-21. doi: 10.3934/bdia.2017020
    [5] Amanda Working, Mohammed Alqawba, Norou Diawara, Ling Li . TIME DEPENDENT ATTRIBUTE-LEVEL BEST WORST DISCRETE CHOICE MODELLING. Big Data and Information Analytics, 2018, 3(1): 55-72. doi: 10.3934/bdia.2018010
    [6] Xiaoxiao Yuan, Jing Liu, Xingxing Hao . A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems. Big Data and Information Analytics, 2017, 2(1): 39-58. doi: 10.3934/bdia.2017007
    [7] Yaguang Huangfu, Guanqing Liang, Jiannong Cao . MatrixMap: Programming abstraction and implementation of matrix computation for big data analytics. Big Data and Information Analytics, 2016, 1(4): 349-376. doi: 10.3934/bdia.2016015
    [8] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong . An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data and Information Analytics, 2017, 2(1): 23-37. doi: 10.3934/bdia.2017006
    [9] Wenxue Huang, Qitian Qiu . Forward Supervised Discretization for Multivariate with Categorical Responses. Big Data and Information Analytics, 2016, 1(2): 217-225. doi: 10.3934/bdia.2016005
    [10] Yiwen Tao, Zhenqiang Zhang, Bengbeng Wang, Jingli Ren . Motality prediction of ICU rheumatic heart disease with imbalanced data based on machine learning. Big Data and Information Analytics, 2024, 8(0): 43-64. doi: 10.3934/bdia.2024003
  • Tumor necrosis factor (TNF) is the name giving member of a large cytokine family mirrored by a respective cell membrane receptor super family. TNF itself is a strong proinflammatory regulator of the innate immune system, but has been also recognized as a major factor in progression of autoimmune diseases. A subgroup of the TNF ligand family, including TNF, signals via so-called death receptors, capable to induce a major form of programmed cell death, called apoptosis. Typical for most members of the whole family, death ligands form homotrimeric proteins, capable to bind up to three of their respective receptor molecules. But also unligated receptors occur on the cell surface as homomultimers due to a homophilic interaction domain. Based on these two interaction motivs (ligand/receptor and receptor/receptor) formation of large ligand/receptor clusters can be postulated which have been also observed experimentally. We use here a mass action kinetics approach to establish an ordinary differential equations model describing the dynamics of primary ligand/receptor complex formation as a basis for further clustering on the cell membrane. Based on available experimental data we develop our model in a way that not only ligand/receptor, but also homophilic receptor interaction is encompassed. The model allows formation of two distict primary ligand/receptor complexes in a ligand concentration dependent manner. At extremely high ligand concentrations the system is dominated by ligated receptor homodimers.


    1. Introduction

    Multi-nominal data are common in scientific and engineering research such as biomedical research, customer behavior analysis, network analysis, search engine marketing optimization, web mining etc. When the response variable has more than two levels, the principle of mode-based or distribution-based proportional prediction can be used to construct nonparametric nominal association measure. For example, Goodman and Kruskal [3,4] and others proposed some local-to-global association measures towards optimal predictions. Both Monte Carlo and discrete Markov chain methods are conceptually based on the proportional associations. The association matrix, association vector and association measure were proposed by the thought of proportional associations in [9]. If there is no ordering to the response variable's categories, or the ordering is not of interest, they will be regarded as nominal in the proportional prediction model and the other association statistics.

    But in reality, different categories in the same response variable often are of different values, sometimes much different. When selecting a model or selecting explanatory variables, we want to choose the ones that can enhance the total revenue, not just the accuracy rate. Similarly, when the explanatory variables with cost weight vector, they should be considered in the model too. The association measure in [9], $\omega^{Y|X}$, doesn't consider the revenue weight vector in the response variable, nor the cost weight in the explanatory variables, which may lead to less profit in total. Thus certain adjustments must be made for a better decisionning.

    To implement the previous adjustments, we need the following assumptions:

    $\bullet$ $X$ and $Y$ are both multi-categorical variables where $X$ is the explanatory variable with domain $\{ 1, 2, ..., \alpha\}$ and $Y$ is the response variable with domain $\{ 1, 2, ..., \beta\}$ respectively;

    $\bullet$ the amount of data collected in this article is large enough to represent the real distribution;

    $\bullet$ the model in the article mainly is based on the proportional prediction;

    $\bullet$ the relationship between $X$ and $Y$ is asymmetric;

    It needs to be addressed that the second assumption is probably not always the case. The law of large number suggests that the larger the sample size is, the closer the expected value of a distribution is to the real value. The study of this subject has been conducted for hundreds of years including how large the sample size is enough to simulate the real distribution. Yet it is not the major subject of this article. The purpose of this assumption is nothing but a simplification to a more complicated discussion.

    The article is organized as follows. Section 2 discusses the adjustment to the association measure when the response variable has a revenue weight; section 3 considers the case where both the explanatory and the response variable have weights; how the adjusted measure changes the existing feature selection framework is presented in section 4. Conclusion and future works will be briefly discussed in the last section.


    2. Response variable with revenue weight vector

    Let's first recall the association matrix $\{\gamma^{s, t}(Y|X)\}$ and the association measure $\omega^{Y|X}$ and $\tau^{Y|X}$.

    $ γs,t(Y|X)=E(p(Y=s|X)p(Y=t|X))p(Y=s)=αi=1p(X=i|Y=s)p(Y=t|X=i);s,t=1,2,..,βτY|X=ωY|XEp(Y)1Ep(Y)ωY|X=EX(EY(p(Y|X)))=βs=1αi=1p(Y=s|X=i)2p(X=i)=βs=1γssp(Y=s)
    $
    (1)

    $\gamma^{st}(Y|X)$ is the $(s, t)$-entry of the association matrix $\gamma(Y|X)$ representing the probability of assigning or predicting $Y=t$ while the true value is in fact $Y=s$. Given a representative train set, the diagonal entries, $\gamma^{ss}$, are the expected accuracy rates while the off-diagonal entries of each row are the expected first type error rates. $\omega^{Y|X}$ is the association measure from the explanatory variable $X$ to the response variable $Y$ without a standardization. Further discussions to these metrics can be found in [9].

    Our discussion begins with only one response variable with revenue weight and one explanatory variable without cost weight. Let $R=(r_1, r_2, ..., r_{\beta})$ to be the revenue weight vector where $r_{s}$ is the possible revenue for $Y=s$. A model with highest revenue in total is then the ideal solution in reality, not just a model with highest accuracy. Therefore comes the extended form of $\omega^{Y|X}$ with weight in $Y$ as in 2:

    Definition 2.1.

    $ ˆωY|X=βs=1αi=1p(Y=s|X=i)2rsp(X=i)=βs=1γssp(Y=s)rsrs>0,s=1,2,3...,β
    $
    (2)

    Please note that $\omega^{Y|X}$ is equivalent to $\tau^{Y|X}$ for given $X$ and $Y$ in a given data set. Thus the statistics of $\tau^{Y|X}$ will not be discussed in this article.

    It is easy to see that $\widehat{\omega}^{Y|X}$ is the expected total revenue for correctly predicting $Y$. Therefore one explanatory variable $X_1$ with $\widehat{\omega}^{Y|X_1}$ is preferred than another $X_2$ if $\widehat{\omega}^{Y|X_1}\geq \widehat{\omega}^{Y|X_2}$. It is worth mentioning that $\widehat{\omega}^{Y|X}$ is asymmetric, i.e., $\widehat{\omega}^{Y|X}\neq\widehat{\omega}^{X|Y}$ and that $\omega^{Y|X}=\widehat{\omega}^{Y|X}$ if $r_1=r_2=...=r_{\beta}=1$.

    Example.Consider a simulated data motivated by a real situation. Suppose that variable $Y$ is the response variable indicating the different computer brands that the customers bought; $X_1$, as one explanatory variable, shows the customers' career and $X_2$, as another explanatory variable, tells the customers' age group. We want to find a better explanatory variable to generate higher revenue by correctly predicting the purchased computer's brand. We further assume that $X_1$ and $X_2$ both contain 5 categories, $Y$ has 4 brands and the total number of rows is $9150$. The contingency table is presented in 1.

    Table 1. Contingency tables:$X_1$ vs $Y$ and $X_2$ vs $Y$.
    $X_1|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $X_2|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$
    $x_{1_1}$ 1000 100 500 400 $x_{2_1}$ 500 300 200 1500
    $x_{1_2}$ 200 1500 500 300 $x_{2_2}$ 500 400 400 50
    $x_{1_3}$ 400 50 500 500 $x_{2_3}$ 500 500 300 700
    $x_{1_4}$ 300 700 500 400 $x_{2_4}$ 500 400 1000 100
    $x_{1_5}$ 200 500 400 200 $x_{2_5}$ 200 400 500 200
     | Show Table
    DownLoad: CSV

    Let us first consider the association matrix $\{\gamma^{Y|X}\}$. Predicting $Y$ with the information of $X_1$, or $X_2$ is given by the association matrix $\gamma(Y|X_1)$, or $\gamma(Y|X_2)$ as in Table 2.

    Table 2. Association matrices:$X_1$ vs $Y$ and $X_2$ vs $Y$.
    $Y|\hat{Y}$ $\hat{y_1}|X_1$ $\hat{y_2}|X_1$ $\hat{y_3}|X_1$ $\hat{y_4}|X_1$ $Y|\hat{Y}$ $\hat{y_1}|X_2$ $\hat{y_2}|X_2$ $\hat{y_3}|X_2$ $\hat{y_4}X_2$
    $y_1$ 0.34 0.18 0.27 0.22 $y_1$ 0.26 0.22 0.27 0.25
    $y_2$ 0.13 0.48 0.24 0.15 $y_2$ 0.25 0.24 0.29 0.23
    $y_{3}$ 0.24 0.28 0.27 0.21 $y_{3}$ 0.25 0.24 0.36 0.15
    $y_{4}$ 0.25 0.25 0.28 0.22 $y_{4}$ 0.22 0.18 0.14 0.46
     | Show Table
    DownLoad: CSV

    Please note that $Y$ contains the true values while $\hat{Y}$ is the guessed one. One can see from this table that the accuracy rate of predicting $y_1$ and $y_2$ by $X_1$ on the left are larger than that on the right. The cases of $y_{3}$ and $y_{4}$, on the other hand, are opposite.

    The correct prediction contingency tables of $X_1$ and $Y$, denoted as $W_1$, plus that of $X_2$ and $Y$, denoted as $W_2$, can be simulated through Monte Carlo simulation as in Table 3.

    Table 3. Contingency table for correct predictions: $W_1$ and $W_2$.
    $X_1|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $X_2|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$
    $x_{1_1}$ 471 6 121 83 $x_{2_1}$ 98 34 19 926
    $x_{1_2}$ 101 746 159 107 $x_{2_2}$ 177 114 113 1
    $x_{1_3}$ 130 1 167 157 $x_{2_3}$ 114 124 42 256
    $x_{1_4}$ 44 243 145 85 $x_{2_4}$ 109 81 489 6
    $x_{1_5}$ 21 210 114 32 $x_{2_5}$ 36 119 206 28
     | Show Table
    DownLoad: CSV

    The total number of the correct predictions by $X_1$ is $3142$ while it is $3092$ by $X_2$, meaning the model with $X_1$ is better than $X_2$ in terms of accurate prediction. But it maybe not the case if each target class has different revenues. Assuming the revenue weight vector of $Y$ is $R=(1, 1, 2, 2)$, we have the association measure of $\omega^{Y|X}$, and $\widehat{\omega}^{Y|X}$ as in Table 4:

    Table 4. Association measures: $\omega^{Y|X}$, and $\widehat{\omega}^{Y|X}$.
    $X $ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ total revenue average revenue
    $X_1$ 0.3406 0.456 4313 0.4714
    $X_2$ 0.3391 0.564 5178 0.5659
     | Show Table
    DownLoad: CSV

    Given that $revenue=\sum_{i, s}W_{k}^{i, s}r_{s}, i=1, 2, ..., \alpha, s=1, 2, ..., \beta, k=1, 2$, we have the revenue for $W_1$ as $4313$, and that for $W_2$ as $5178$. Divide the revenue by the total sample size, 9150, we can obtain $0.4714$ and $0.5659$ respectively. Contrasting these to $\widehat{\omega}^{Y|X_1}$ and $\widehat{\omega}^{Y|X_2}$ above, we believe that they are similar, which means then $\widehat{\omega}^{Y|X}$ is truly the expected revenue.

    In summary, it is possible for an explanatory variable $X$ with bigger $\widehat{\omega}^{Y|X}$, i.e, the larger revenue, but with smaller $\omega^{Y|X}$, i.e., the smaller association. When the total revenue is of the interest, it should be the better variable to be selected, not the one with larger association.


    3. Explanatory variable with cost weight and response variable with revenue weight

    Let us further discuss the case with cost weight vector in predictors in addition to the revenue weight vector in the dependent variable. The goal is to find a predictor with bigger profit in total. We hence define the new association measure as in 3.

    Definition 3.1.

    $ \bar{\omega}^{Y|X}=\sum\limits_{i=1}^{\alpha}\sum\limits_{s=1}^{\beta}p(Y=s|X=i)^2\frac{r_{s}}{c_{i}}p(X=i)\\ $ (3)

    $c_{i}>0, i=1, 2, 3, ..., \alpha, $ and $r_{s}>0, s=1, 2, ..., \beta.\notag\\$

    $c_{i}$ indicates the cost weight of the $i$th category in the predictor and $r_{s}$ means the same as in the previous section. $\bar{\omega}^{Y|X}$ is then the expected ratio of revenue and cost, namely RoI. Thus a larger $\bar{\omega}^{Y|X}$ means a bigger profit in total. A better variable to be selected then is the one with bigger $\bar{\omega}^{Y|X}$. We can see that $\bar{\omega}^{Y|X}$ is an asymmetric measure, meaning $\bar{\omega}^{Y|X}\neq\bar{\omega}^{Y|X}$. When $c_1=c_2=...=c_{\alpha}=1$, Equation 3 is exactly Equation 2; when $c_1=c_2=...=c_{\alpha}=1$ and $r_1=r_2=...=r_{\beta}=1$, equation 3 becomes the original equation 1.

    Example. We first continue the example in the previous section with new cost weight vectors for $X_1$ and $X_2$ respectively. Assuming $C_1=(0.5, 0.4, 0.3, 0.2, 0.1)$, $C_2=(0.1, 0.2, 0.3, 0.4, 0.5)$ and $R=(1, 1, 1, 1)$, we have the associations in Table 5.

    Table 5. Association with/without cost vectors: $X_1$ and $X_2$.
    $X $ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit
    $X_1$ 0.3406 0.3406 1.3057 12016.17 1.3132
    $X_2$ 0.3391 0.3391 1.8546 17072.17 1.8658
     | Show Table
    DownLoad: CSV

    By $profit=\sum_{i, s}W_{k}^{i, s}\frac{r_{s}}{C_{k_{i}}}, i=1, 2, .., \alpha;s=1, 2, .., \beta$ and $k=1, 2$ where $W_{k}$ is the corresponding prediction contingency table, we have the profit for $X_1$ as $12016.17$ and that of $X_2$ as $17072.17$. When both divided by the total sample size 9150, they change to 1.3132 and 1.8658, similar to $\bar{\omega}(Y|X_1)$ and $\bar{\omega}(Y|X_2)$. It indicates that $\bar{\omega}^{Y|X}$ is the expected RoI. In this example, $X_2$ is the better variable given the cost and the revenue vectors are of interest.

    We then investigate how the change of cost weight affect the result. Suppose the new weight vectors are: $R=(1, 1, 1, 1)$, $C_1=(0.1, 0.2, 0.3, 0.4, 0.5)$ and $C_2=(0.5, 0.4, 0.3, 0.2, 0.1)$, we have the new associations in Table 6.

    Table 6. Association with/without new cost vectors: $X_1$ and $X_2$.
    $X $ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit
    $X_1$ 0.3406 0.3406 1.7420 15938.17 1.7419
    $X_2$ 0.3391 0.3391 1.3424 12268.17 1.3408
     | Show Table
    DownLoad: CSV

    Hence $\bar{\omega}^{Y|X_1}>\bar{\omega}^{Y|X_2}$, on the contrary to the example with the old weight vectors. Thus the right amount of weight is critical to define the better variable regarding the profit in total.


    4. The impact on feature selection

    By the updated association defined in the previous section, we present the feature selection result in this section to a given data set $S$ with explanatory categorical variables $V_1, V_2, .., V_{n}$ and a response variable $Y$. The feature selection steps can be found in [9].

    At first, consider a synthetic data set simulating the contribution factors to the sales of certain commodity. In general, lots of factors could contribute differently to the commodity sales: age, career, time, income, personal preference, credit, etc. Each factor could have different cost vectors, each class in a variable could have different cost as well. For example, collecting income information might be more difficult than to know the customer's career; determining a dinner waitress' purchase preference is easier than that of a high income lawyer. Therefore we just assume that there are four potential predictors, $V_1, V_2, V_{3}, V_{4}$ within the data set with a sample size of 10000 and get a feature selection result by monte carlo simulation in Table 7.

    Table 7. Simulated feature selection: one variable.
    $X$ $|Dmn(X)|$ $\omega^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit
    $V_1$ 7 0.3906 3.5381 35390 3.5390
    $V_2$ 4 0.3882 3.8433 38771 3.8771
    $V_{3}$ 4 0.3250 4.8986 48678 4.8678
    $V_{4}$ 8 0.3274 3.7050 36889 3.6889
     | Show Table
    DownLoad: CSV

    The first variable to be selected is $V_1$ using $\omega^{Y|X}$ as the criteria according to [9]. But it is $V_{3}$ that needs to be selected as previously discussed if the total profit is of interest. Further we assume that the two variable combinations satisfy the numbers in Table 8 by, again, monte carlo simulation.

    Table 8. Simulated feature selection: two variables.
    $X_1, X_2$ $|Dmn(X_1, X_2)|$ $\omega^{Y|(X_1, X_2)}$ $\bar{\omega}^{Y|(X_1, X_2)}$ total profit average profit
    $V_1,V_2$ 28 0.4367 1.8682 18971 1.8971
    $V_1, V_{3}$ 28 0.4025 2.1106 20746 2.0746
    $V_1, V_{4}$ 56 0.4055 1.8055 17915 1.7915
    $V_{3}, V_2$ 16 0.4055 2.3585 24404 2.4404
    $V_{3}, V_{4}$ 32 0.3385 2.0145 19903 1.9903
     | Show Table
    DownLoad: CSV

    As we can see, all $\omega^{Y|(X_1, X_2)}\geq\omega^{Y|X_1}$, but it is not case for $\bar{\omega}^{Y|(X_1, X_2)}$ since the cost gets larger with two variables thus the profit drops down. As in one variable scenario, the better two variable combination with respect to $\omega^{Y|(X_1, X_2)}$ is $(V_1, V_2)$ while $\bar{\omega}^{Y|(X_1, X_2)}$ suggests $(V_{3}$, $V_2)$ is the better choice.

    In summary, the updated association with cost and revenue vector not only changes the feature selection result by different profit expectations, it also reflects a practical reality that collecting information for more variables costs more thus reduces the overall profit, meaning more variables is not necessarily better on a Return-Over-Invest basis.


    5. Conclusions and remarks

    We propose a new metrics, $\bar{\omega^{Y|X}}$ in this article to improve the proportional prediction based association measure, $\omega^{Y|X}$, to analyze the cost and revenue factors in the categorical data. It provides a description to the global-to-global association with practical RoI concerns, especially in a case where response variables are multi-categorical.

    The presented framework can also be applied to high dimensional cases as in national survey, misclassification costs, association matrix and association vector [9]. It should be more helpful to identify the predictors' quality with various response variables.

    Given the distinct character of this new statistics, we believe it brings us more opportunities to further studies of finding the better decision for categorical data. We are currently investigating the asymptotic properties of the proposed measures and it also can be extended to symmetrical situation. Of course, the synthetical nature of the experiments in this article brings also the question of how it affects a real data set/application. It is also arguable that the improvements introduced by the new measures probably come from the randomness. Thus we can use $k$-fold cross-validation method to better support our argument in the future.


  • Reader Comments
  • © 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3116) PDF downloads(545) Cited by(4)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog