A method for predicting epithelial cell gene regulation relationships based on directed graph convolution.

Abstract

This invention discloses a method for predicting gene regulatory relationships in epithelial cells based on directed graph convolution, comprising: 1. obtaining the expression values ​​of epithelial cell genes and preprocessing the feature values; 2. obtaining epithelial cell gene sequence data and inputting the gene sequence data into a bidirectional gated recurrent unit of a recurrent neural network model to obtain the sequence features of this type of gene; 3. calculating biological features for the epithelial cell gene sequences using biologically defined formulas; 4. concatenating the features obtained in the previous steps and inputting them into a directed graph convolutional neural network to train an epithelial cell gene regulatory network and its output prediction score matrix, thereby determining whether a regulatory relationship exists between genes based on the relationship between the prediction score matrix and a threshold. This invention can accurately predict the control relationships between genes in human epithelial cells, helping researchers to study the nature of organisms more efficiently.

Description

Technical Field

[0001] This invention relates to the fields of intelligent information processing and bioinformatics, specifically a method for predicting gene regulatory relationships in epithelial cells based on directed graph convolution. Background Technology

[0002] Transcriptional regulation is crucial for cells, allowing them to adapt to changing environments and dynamically regulate gene expression in response to external stimuli. Transcription factors (TFs) exert their effects by regulating themselves and their corresponding target genes. Transcription factors, target genes, and their interactions constitute gene regulatory networks (GRNs). GRNs aim to capture the dependencies between these molecule entities, transcription factors and target genes. Gene regulatory networks play a vital role in the growth and development of organisms by controlling gene expression. Specifically, GRNs are core elements in maintaining life processes, controlling biochemical reactions, and regulating compound levels, playing important roles in various organisms and systems. Therefore, accurately constructing GRNs can improve our understanding of gene regulation mechanisms and the molecular mechanisms of organisms.

[0003] To better understand the regulatory mechanisms between genes, many algorithms have been developed in recent years for constructing Gene-Recurrent Neural Networks (GRNs). Information-theoretic methods quantify the statistical dependencies between genes based on their correlations, predicting the GRN structure from the resulting correlation matrix. Boolean network-based methods use Boolean variables to represent gene states and then use Boolean functions to calculate the relationships between genes, capturing dynamic interactions between them. Feature selection is also increasingly being applied to GRN inference, such as GENIE3, dynGENIE3, Jump3, and BiXGBoost. With the development of deep learning, more and more deep learning methods are being applied to GRN inference, such as CNN, TDL, and DeepDRIM.

[0004] The methods described above are all based on traditional approaches such as information theory and traditional machine learning, or they use basic convolutional neural networks or a combination of convolutional neural networks and recurrent neural networks. For directed graph data like GRNs, graph neural networks can be more efficient and convenient, but their use is limited due to the more cumbersome data construction process. Secondly, methods that only use gene expression data tend to have poor prediction results because they cannot uncover deeper information. Furthermore, the large number of low-density gene nodes in GRNs prevents previous models from fully expressing their genetic material. Summary of the Invention

[0005] The present invention addresses the shortcomings of the existing technology by proposing a method for predicting gene regulatory relationships in epithelial cells based on directed graph convolution. This method aims to predict the regulatory relationships between genes more efficiently and accurately, thereby helping biological researchers to study the regulatory relationships between genes more effectively.

[0006] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0007] The present invention provides a method for predicting epithelial cell gene regulation relationships based on directed graph convolution, characterized by comprising the following steps:

[0008] Step 1: Obtain the gene set X of epithelial cells and preprocess it to form an n×c dimensional gene expression feature matrix F1, where n is the number of genes and c is the dimension of gene expression value; take each gene in the epithelial cell as a gene node and the control relationship between each gene as the edge between gene nodes to construct the gene control graph G of epithelial cells.

[0009] Step 2: Construct a conditional variational autoencoder and use it to process the features of gene nodes and their neighboring gene nodes in the gene control graph G of epithelial cells to obtain an n×h dimensional neighborhood feature matrix F2, where h represents the neighborhood feature dimension of each gene node.

[0010] Step 3: Obtain the n×t dimensional gene sequence matrix S of epithelial cells and input it into a bidirectional gated recurrent unit for processing to obtain the n×t dimensional sequence feature matrix F3; t represents the dimension of the sequence feature of each gene node.

[0011] The n×t-dimensional gene sequence matrix S is calculated using biological formulas to obtain the n×e-dimensional biofeature matrix F4;

[0012] Step 4: Concatenate F1, F2, F3 and F4 to obtain an n×(c+h+t+e) dimensional gene feature matrix F5;

[0013] Step 5: Construct a directed graph convolutional neural network and use it to process F5 to obtain a prediction score matrix showing the regulatory relationships between all gene nodes in the gene control graph G.

[0014] Step 6: Train the directed graph convolutional neural network using stochastic gradient descent and calculate the cross-entropy loss function to update the network parameters until the cross-entropy loss function converges, thereby obtaining the trained epithelial cell gene regulation model.

[0015] Step 7: Input the gene expression feature matrix and gene sequence matrix of epithelial cells into the epithelial cell gene regulation model for prediction, and output the prediction score of the regulatory relationship between genes. Finally, determine whether a regulatory relationship exists by using the set threshold.

[0016] The method for predicting epithelial cell gene regulation relationships based on directed graph convolution described in this invention is also characterized in that the conditional variational autoencoder comprises two parts: an encoder network and a decoder network, wherein both the encoder network and the decoder network are composed of multilayer sensing mechanisms.

[0017] The characteristics of any gene node q in the gene control graph G of epithelial cells are i q As conditional information for the conditional variational autoencoder, the features i of the neighboring gene nodes l of gene node q are used. q，l As input data to the conditional variational autoencoder, the conditional variational autoencoder uses equation (1) to obtain a 1×h-dimensional neighborhood feature vector y. q Thus, the neighborhood feature vectors of all n gene nodes in the gene control graph G are concatenated to obtain an n×h dimensional neighborhood feature matrix F2, where h represents the neighborhood feature dimension of each gene node.

[0018] y q =f(W1z) q +W2c+b) (1)

[0019] In equation (1), f is the activation function, W1 and W2 are two weight matrices in the decoder network, b is the bias term in the decoder network, and z q Let be a latent variable of gene node q, and have:

[0020] z q =NN μ (i q，l i q )+exp(NN σ (i q，l i q (2)

[0021] In equation (2), NN μ and NN σ This represents the code used to learn z in the encoder network. q The mean and standard deviation of the distribution.

[0022] Step 5 includes:

[0023] Step 5.1: Obtain the degree information of gene node q in the gene control graph G using equations (3), (4), and (5), including the first-order degree information A1, the second-order in-degree information A2, and the second-order out-degree information A3 of node q:

[0024] A1=A(q,a) (3)

[0025]

[0026]

[0027] In equation (3), A(q, a) represents the adjacency matrix of gene node a in the gene control graph G that has a control relationship with gene node q;

[0028] In equation (4), P k，q P represents the edge from gene node k to gene node q in the gene control graph G. k，b P represents the edge from gene node k to gene node b in the gene control graph G; k，v This represents the edge in the gene control graph G from gene node k to gene node v;

[0029] In equation (5), P q，k P represents the edge in the gene control graph G from gene q to gene k. d，k This indicates that gene d in gene control graph G has an edge pointing to gene k; P v，k This represents the edge in the gene control graph G from gene v to gene k;

[0030] Step 5.2: Construct the first-order in-degree matrix A of the gene control graph G from the degree information of all gene nodes in the gene control graph G. F Second-order in-degree matrix and second-order out-degree matrix

[0031] Step 5.3: Using equations (6), (7), and (8), obtain the potential regulatory relationships between each gene node, including: the potential relationship Z of first-order information. F Potential relationships of second-order in-degree information The potential relationship with second-order out-degree information

[0032]

[0033]

[0034]

[0035] In equation (6), Denotes a self-loop matrix, and It is The diagonalized matrix, where Θ is the filter parameter matrix and I is the identity matrix;

[0036] In equation (7), Let represent a self-loop matrix with second-order in-degree, and It is The diagonalized matrix;

[0037] In equation (8), Let represent a self-loop matrix with second-order out-degree, and It is The diagonalized matrix;

[0038] Step 5.4: Use equation (9) to obtain the prediction score matrix R of the regulatory relationships between all gene nodes in the gene control graph G:

[0039]

[0040] In equation (9), α and β are two weight parameters, Concat is a cascade operation, and Softmax is an activation function.

[0041] The present invention provides an electronic device, including a memory and a processor, wherein the memory is used to store a program that supports the processor in executing the epithelial cell gene regulation relationship prediction method based on directed graph convolution, and the processor is configured to execute the program stored in the memory.

[0042] The present invention discloses a computer-readable storage medium on which a computer program is stored, wherein the computer program, when executed by a processor, performs the steps of the method for predicting epithelial cell gene regulation relationships based on directed graph convolution.

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0044] 1. GRN contains a large number of low-degree genes, which can affect the expression performance of existing neural network models and make it difficult for the models to efficiently predict the regulatory relationships between genes. Therefore, this invention uses a conditional variational autoencoder (CVAE) to enhance the local features of nodes in the graph neural network, thereby enhancing the expressive power of the graph neural network and solving the problem of difficulty in mining the association information of low-degree gene nodes in the gene regulatory network.

[0045] 2. Current neural network models only use the expression values ​​of cellular genes as the feature matrix input to the neural network model. However, existing deep learning computational methods struggle to extract more useful information from gene expression data. Therefore, this invention uses recurrent neural networks to extract gene sequence feature values ​​and employs biologically meaningful sequence feature descriptors to extract gene feature values ​​from different perspectives, thereby improving the predictive performance of the epithelial cell gene regulation model.

[0046] 3. Most neural network-based methods use traditional convolutional neural networks or a combination of convolutional neural networks and recurrent neural networks. Furthermore, these methods predefine the direction of gene regulation from the regulatory factor to the target gene, which can lead to the loss of some circular regulatory relationships. Therefore, this invention uses a directed graph convolutional neural network to efficiently process directed graph data while more accurately predicting gene regulatory relationships. Attached Figure Description

[0047] Figure 1 This is a flowchart of the method of the present invention;

[0048] Figure 2 This is a structural diagram of the conditional variational autoencoder of the present invention;

[0049] Figure 3 The plots show the area under the receiver operating characteristic (AUC) and area under the precision-recall curves of each method under ten runs of five-fold cross-validation in five different networks in the DREAM4 simulation data, compared with other existing methods.

[0050] Figure 4 The plots show the area under the receiver operating characteristic (AUC) and area under the precision-recall curve (ARBC) of each method under ten five-fold cross-validation runs in three different networks using real E. coli data, compared with other existing methods.

[0051] Figure 5 This is a comparison of the area under the receiver operating characteristic (AUC) and area under the precise recall (AUC) plots of the present invention in abnormal bladder urothelial cells with other existing methods. Detailed Implementation

[0052] In this embodiment, a gene regulatory relationship prediction method based on directed graph convolution is described, such as... Figure 1 As shown, the procedure is as follows:

[0053] Step 1: Obtain the gene set X of epithelial cells and preprocess it to form an n×c dimensional gene expression feature matrix F1, where n is the number of genes and c is the dimension of gene expression values. Treat each gene in the epithelial cell as a node, and the control relationships between genes as edges between nodes, thereby constructing the gene control graph G of the epithelial cells.

[0054] Step 2: Construct a conditional variational autoencoder, consisting of two parts: an encoder network and a decoder network, such as... Figure 2 As shown. Both the encoder and decoder networks are composed of multilayer sensing mechanisms;

[0055] Step 2.1: Identify the characteristics of any gene node q in the gene control graph G of epithelial cells. qAs conditional information for the conditional variational autoencoder, the features i of the neighboring gene nodes l of gene node q are used. q，l As input data to the conditional variational autoencoder, the conditional variational autoencoder uses equation (1) to obtain a 1×h-dimensional neighborhood feature vector y. q Thus, the neighborhood feature vectors of all n gene nodes in the gene control graph G are concatenated to obtain an n×h dimensional neighborhood feature matrix F2.

[0056] y q =f(W1z) q +W2c+b) (1)

[0057] In equation (1), f is the activation function, W1 and W2 are two weight matrices in the decoder network, b is the bias term in the decoder network, and z q Let be a latent variable of gene node q, and have:

[0058] z q =NN μ (i q，l i q )+exp(NN σ (i q，l i q (2)

[0059] In equation (2), NN μ and NN σ This represents the code used to learn z in the encoder network. q The mean and standard deviation of the distribution.

[0060] like Figure 2 As shown, the encoder and decoder structures of the conditional variational autoencoder both consist of two-layer multilayer perceptrons. The encoder uses the features i of the gene node q. q (Right now Figure 2 Xv) is the input conditional information, and the features i of the neighboring gene nodes l of gene node q are used. q，l (Right now Figure 2 In this context, Xu represents the data input to the encoder. The encoder network learns the mean and standard deviation of the distribution of the input data (i.e., the neighboring gene nodes l of gene node q). Then, z is sampled from the distribution N(μ, σ) obtained from the encoder. q This is used as the input data for the decoder, while simultaneously using the features i of the gene node q. q Given the input conditional information, the final result is a 1×h-dimensional neighborhood feature vector y. q After concatenating the neighborhood feature vectors of all n gene nodes in the gene control graph G, an n×h dimensional neighborhood feature matrix F2 is obtained.

[0061] Step 3: Obtain the n×t dimensional gene sequence matrix S of epithelial cells and input it into a bidirectional gated loop unit for processing to obtain the n×t dimensional sequence feature matrix F3;

[0062] The gene sequence matrix S is the base sequence of the gene (such as ATCGGCT...). The base sequence length of each gene is not the same. In order to process the gene base sequence through a neural network (bidirectional gated recurrent unit), we pad the shorter sequences with zeros to make them the same length as the longest gene sequence. This allows the sequences of all genes to be concatenated into a matrix (i.e., sequence matrix S) and input into the neural network for processing.

[0063] The n×t-dimensional gene sequence matrix S is calculated using biological formulas to obtain the n×e-dimensional biofeature matrix F4;

[0064] Biological formulas can be obtained through collection, meaning that the biological features currently input to the model are not unique. Any number of biological features can be added, and the more such formulas are collected, the more effective the biological meaning will be, and the better the final prediction results will be.

[0065] Step 4: Concatenate F1, F2, F3 and F4 to obtain an n×(c+h+t+e) dimensional gene feature matrix F5;

[0066] Step 5: Construct a directed graph convolutional neural network;

[0067] Step 5.1: Obtain the degree information of gene node q in the gene control graph G using equations (3), (4), and (5), including the first-order degree information A1, the second-order in-degree information A2, and the second-order out-degree information A3 of node q:

[0068] A1=A(q,a) (3)

[0069]

[0070]

[0071] In equation (3), A is the adjacency matrix of the gene control graph G, a represents a gene in the gene control graph G that has a control relationship with gene q; A(q, a) represents the adjacency matrix of gene node a in the gene control graph G that has a control relationship with gene node q.

[0072] In equation (4), P k，q P indicates that there exists an edge in the gene control graph G pointing from gene k to gene q. k，b This indicates that there exists an edge in the gene control graph G that points from gene k to gene b; This represents the number of edges in the gene control graph G that point from gene k to all other genes.

[0073] In equation (5), P q，k P indicates that there exists an edge in the gene control graph G pointing from gene q to gene k. d，k This indicates that in the gene control graph G, there exists a gene d with an edge pointing to gene k; This represents the number of edges in the gene control graph G from all other genes to gene k.

[0074] In the formula for calculating the second-order in-degree information, the numerator represents gene k that points to both gene q and gene b, and the denominator represents all gene nodes that gene k points to. The final result of the formula is the proportion of a node that points to both gene q and gene b out of the total number of nodes it points to.

[0075] Step 5.2: Organize the node degree information of all genes in the gene control graph G into a matrix to obtain the first-order in-degree matrix A of the nodes in the gene control graph G. F Second-order in-degree matrix and second-order out-degree matrix

[0076] Step 5.3: Use equations (6), (7), and (8) to obtain the potential regulatory relationships between various gene nodes, including: the potential relationship Z through first-order degree information. F Potential relationships of second-order in-degree information The potential relationship with second-order out-degree information

[0077]

[0078]

[0079]

[0080] In equation (6), Denotes a self-loop matrix, and It is The diagonalized matrix, where Θ is the filter parameter matrix and I is the identity matrix;

[0081] In equation (7), Let represent a self-loop matrix with second-order in-degree, and It is The diagonalized matrix;

[0082] In equation (8), Let represent a self-loop matrix with second-order out-degree, and It is The diagonalized matrix;

[0083] The purpose of using a self-loop matrix is ​​to eliminate the influence of undirected graph structure edges when aggregating neighbor node information. Without adding self-loops, nodes would not be able to aggregate their own feature information because the edges of an undirected graph do not have a direction.

[0084] Step 5.4: Use equation (9) to obtain the prediction score matrix R, which represents the probability of associations between genes in the gene control graph G:

[0085]

[0086] In equation (9), α and β are two weight parameters, Concat is a cascade operation, and Softmax is an activation function;

[0087] In the prediction score matrix R, if the prediction score indicating a correlation between any two nodes is greater than the set threshold, it indicates that there is a regulatory relationship between the two genes and the direction of regulation is from the regulator to the target gene; otherwise, it indicates that there is no regulatory relationship between the two genes.

[0088] Step 6: Train the directed graph convolutional neural network using stochastic gradient descent and calculate the cross-entropy loss function to update the model's hyperparameters until the cross-entropy loss function converges. Simultaneously, continuously adjust the set threshold to obtain a well-trained epithelial cell gene regulation model.

[0089] Step 7: Input the gene and sequence feature information of epithelial cells into the epithelial cell gene regulation model for prediction, and output the prediction score of the regulatory relationship between genes. Finally, determine whether there is a definite regulatory relationship by using the set threshold.

[0090] In this embodiment, an electronic device includes a memory and a processor. The memory stores a program that supports the processor in executing the above-described method, and the processor is configured to execute the program stored in the memory.

[0091] In this embodiment, a computer-readable storage medium stores a computer program, which is executed by a processor to perform the steps of the above method.

[0092] To evaluate the model's performance, the results were compared with individual classifiers and other methods, using the Area Under the Receiver Operating Characteristic (AUROC) as the evaluation metric. AAUROC, or Area Under the Receiver Operating Characteristic, is a value between 0 and 1, providing a direct assessment of the classifier's performance; a higher AUC indicates a better classifier. Ten five-fold cross-validations were used in the experiments, and the average value was calculated as the model's result.

[0093] Figure 3 The results of this application's model on the DREAM4 simulated dataset are presented, comparing it with other prediction models Jump3, dynamical GEne Network Inference with Ensemble of Trees (dynGENIE3), Bidirectional eXtreme Gradient Boosting (BiXGBoost), and Multi-source Multi-model Fusion method (MMFGRN). Experimental results show that the model of this invention outperforms other prediction models on the five network datasets of DREAM4 (networks 1 to 5), verifying the effectiveness and superiority of this model.

[0094] Figure 4 The results of the model presented in this application on a real E. coli dataset are compared with other prediction models Jump3, dynGENIE3, and BiXGBoost. In three network datasets—cold stress, heat stress, and oxygen stress—the model of this invention also outperforms other prediction models on the real dataset, demonstrating its superiority and indicating that it can more accurately predict the regulatory relationships between genes.

[0095] Figure 5This paper presents the implementation results of the proposed model on bladder urothelial abnormal cell data, comparing it with GENIE-Random Forest (GENIE-RF), GENIE-Extrem Tree (GENIE-ET), Network Inference using Multiple Ensemble Feature Importance Algorithms (NIMEFI), GEne Network Inference method based on a Multi-level Strategy (GENIMS), Partial LeastSquares NET (PLSNET), NARROMI, and Gene Networks Inference based on Projection and Lagged Regression (GNIPLR). Experimental results demonstrate that the proposed model outperforms other prediction models in real-world bladder urothelial abnormal cell data, validating the feasibility and effectiveness of the proposed model in practical applications.

Claims

1. A method for predicting gene regulatory relationships in epithelial cells based on directed graph convolution, characterized in that, It includes the following steps: Step 1: Obtain the gene set of epithelial cells And preprocessing is performed to form dimensional gene expression feature matrix , For the number of genes, The gene expression value dimension is used; each gene in the epithelial cell is treated as a gene node, and the control relationships between genes are treated as edges between gene nodes, thus constructing a gene control graph of the epithelial cell. ; Step 2: Construct a conditional variational autoencoder and use it to map the gene control map of epithelial cells. The characteristics of gene nodes and their neighboring gene nodes are processed to obtain... dimensional neighborhood feature matrix , This represents the neighborhood feature dimension of each gene node; Step 3: Obtaining epithelial cells 3D gene sequence matrix The data is then input into a bidirectional gated loop unit for processing to obtain... dimensional sequence feature matrix ; The dimension representing the sequence features of each gene node; Using biological formulas 3D gene sequence matrix Calculations were performed to obtain 3D biomarker matrix ; Step 4: , , and By splicing them together, we get a 3D gene feature matrix ; Step 5: Construct a directed graph convolutional neural network and use it for... The process yields a gene control map. A prediction score matrix showing the regulatory relationships between all gene nodes; Step 6: Train the directed graph convolutional neural network using stochastic gradient descent and calculate the cross-entropy loss function to update the network parameters until the cross-entropy loss function converges, thereby obtaining the trained epithelial cell gene regulation model. Step 7: Input the gene and sequence feature information of epithelial cells into the epithelial cell gene regulation model for prediction, and output the prediction score of the regulatory relationship between genes. Finally, determine whether a regulatory relationship exists by using the set threshold.

2. The method for predicting epithelial cell gene regulation relationships based on directed graph convolution according to claim 1, characterized in that, The conditional variational autoencoder comprises two parts: an encoder network and a decoder network, both of which are composed of multilayer sensing mechanisms. Gene control diagram of epithelial cells Any gene node Features As conditional information for conditional variational autoencoders, gene nodes Neighboring gene nodes Features As input data to the conditional variational autoencoder, the conditional variational autoencoder obtains the following using equation (1): 3D neighborhood feature vector Thus, the gene control map All After concatenating the neighborhood feature vectors of each gene node, we obtain dimensional neighborhood feature matrix ; (1) In equation (1), It is an activation function. and These are two weight matrices in the decoder network. It is the bias term in the decoder network. It is a gene node The latent variables are: (2) In equation (2), and This represents the learning process in the encoder network. The mean and standard deviation of the distribution.

3. The method for predicting epithelial cell gene regulation relationships based on directed graph convolution according to claim 2, characterized in that, Step 5 includes: Step 5.1: Obtain the gene control map using equations (3), (4), and (5) respectively. Middle gene node Degree information, including nodes First-order information Second-order in-degree information and second-order out-degree information : (3) (4) (5) In equation (3), Representing a gene control diagram In and gene nodes gene nodes The adjacency matrix; In equation (4), Representing a gene control diagram Middle gene node Pointing to gene nodes The edge, Representing a gene control diagram Middle gene node Pointing to gene nodes The edge; Representing a gene control diagram Middle gene node Pointing to gene nodes The edge; In equation (5), Representing a gene control diagram Zhonggen Pointing to genes The edge, Representing a gene control diagram Zhonggen There are genes that point to the gene The edge; Representing a gene control diagram Zhonggen Pointing to genes The edge; Step 5.2: Gene control map The degree information of all gene nodes constitutes a gene control graph. first-order in-degree matrix Second-order in-degree matrix and second-order out-degree matrix ; Step 5.3: Using equations (6), (7), and (8), obtain the potential regulatory relationships between various gene nodes, including: potential relationships of first-order information. Potential relationships of second-order in-degree information The potential relationship with second-order out-degree information : (6) (7) (8) In equation (6), Denotes a self-loop matrix, and , It is The diagonalized matrix, This is the filter parameter matrix; It is the identity matrix; In equation (7), Let represent a self-loop matrix with second-order in-degree, and , It is The diagonalized matrix; In equation (8), Let represent a self-loop matrix with second-order out-degree, and , It is The diagonalized matrix; Step 5.4: Obtain the gene control map using equation (9) A prediction score matrix showing the regulatory relationships between all gene nodes. : (9) In equation (9), and There are two weight parameters. It is a cascading operation. It is an activation function.

4. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store programs that support the processor in executing the epithelial cell gene regulation relationship prediction method based on directed graph convolution as described in any of claims 1-3, and the processor is configured to execute the programs stored in the memory.

5. A computer-readable storage medium storing a computer program thereon, characterized in that, When the computer program is run by the processor, it performs the steps of the method for predicting epithelial cell gene regulation relationships based on directed graph convolution as described in any of claims 1-3.

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