Intervention device, intervention method, and program

Abstract

An intervention device 1 for suppressing fluctuation of a network in which nodes having a state quantity are connected includes: a state quantity sample acquisition unit 11 for acquiring a plurality of samples of a state quantity; an estimation unit 12; an intervention unit 13; and a control unit 14. The estimation unit 12 selects an eigenvector related to the eigenvalue of the covariance matrix of the plurality of acquired samples as an estimated value of the dominant right eigenvector. The control unit 14 controls the intervention unit 13 so as to intervene in the network in a direction to suppress a state quantity corresponding to a component of the eigenvector related to the selected eigenvalue.

Description

Intervention device, intervention method and program

[0001] The present invention relates to an intervention device, an intervention method, and a program.

[0002] In natural phenomena (e.g., the onset of disease, the occurrence of earthquakes, climate change, etc.) and social phenomena (e.g., road traffic volume, electricity consumption, fluctuations in foreign exchange and stock prices, etc.), when a system reaches a "critical state" (hereinafter also referred to as a "branch point" or "tipping point"), a sudden transition from a previous stable state to another stable state may occur (see, for example, Non-Patent Documents 1 to 5). For example, in the case of a living organism, when a person who was stably healthy up to a certain point reaches a "pre-critical state" (the state immediately before the critical state), which is a "pre-disease state" (a state in which the person is moving away from a healthy state, but has not yet developed a disease), the state then suddenly transitions to a disease, and the disease state continues. If such a pre-critical state (i.e., a sign of a sudden state transition) could be detected, it would be possible to respond to sudden changes and phenomena in natural and social phenomena, including disease.

[0003] The following description focuses on biological systems (especially biological networks modeled as networks) as target systems. However, it should be understood that the technology disclosed herein is not limited to biological systems, but can be widely applied to natural and social systems whose states change due to some factor, such as crustal systems, meteorological systems, ecosystems, transportation systems, power systems, and financial systems.

[0004] Regarding research into the dynamic mechanisms of disease, it is known that the progression process of disease worsening (e.g., asthma attack, onset of cancer) is modeled as a time-dependent nonlinear dynamic system, and by analyzing this, a relationship between state transitions at branching points and sudden worsening of the disease has been discovered (see, for example, non-patent document 6).

[0005] Against this background, a technology for detecting candidates for dynamic network markers (hereinafter also referred to as "DNMs"), which serve as indicators of the state of a system, has been disclosed (see, for example, Patent Document 1). In the case of a living organism, a DNM is also called a dynamic network biomarker (hereinafter referred to as "DNB"), based on measurement data of multiple factor items obtained by measuring the organism. Furthermore, a technology has been disclosed for detecting a pre-disease state between a healthy state and a diseased state, in which changes in the state of a target factor and the connecting factors that are dynamically directly connected to this factor are regarded as local network entropy, and the pre-disease state is detected based on this local network entropy (see, for example, Patent Document 2). Furthermore, a technology has been disclosed for determining whether a DNB has occurred by storing data obtained by collecting and analyzing biological materials from multiple healthy individuals and adding analytical data of biological materials collected only once from a subject (see, for example, Patent Document 3).

[0006] JP 2014-64515 JP 2014-83194 WO 2018 / 207925

[0007] Jose G. Venegas, Tilo Winkler, Guido Musch, Marcos F. Vidal Melo, Dominick Layfield, Nora Tgavalekos, Alan J. Fischman, Ronald J. Callahan, Giacomo Bellani, and R. Scott Harris,"Self-organized patchiness in asthma as a prelude to catastrophic shifts" Nature 434, Nature Publishing Group, pp. 777-782 (2005)Patrick E. McSharry, Leonard A. Smith, and Lionel Tarassenko,"Prediction of epileptic seizures: are nonlinear methods relevant?" Nature Medicine 9, Nature Publishing Group, pp. 241-242 (2003)Roberto Pastor‐Barriuso, Eliseo Guallar, and Josef Coresh,"Transition models for change-point estimation in logistic regression" Statisticsin Medicine 22(7), Wiley-Blackwell, pp. 1141-1162 (2003)Paek SH et al. "Hearing preservation after gamma knife stereotactic radiosurgery of vestibular schwannoma" Cancer 104, Wiley-Blackwell, pp. 580-590 (2005)Liu, J.K., Rovit, R.L., and Couldwell, W.T., "Pituitary Apoplexy" Seminars in Neurosurgery 12, Thieme, pp. 315-320 (2001)G.Tanaka, K.Tsumoto, S.Tsuji, K.Aihara, “Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer” Physica D: Nonlinear Phenomena Volume 237, Issue 20, 15 October 2008, pp. 2616-2627.

[0008] The techniques disclosed in Patent Documents 1 to 3 can detect signs of a transition from a healthy state to a diseased state. If a pre-disease state is detected in this way, and appropriate early treatment can be administered at this stage (i.e., before the diseased state is reached) to restore the patient to a healthy state, this is expected to have the excellent effect of reducing treatment costs and increasing the chances of recovery. To achieve such early treatment, it is necessary to know what kind of intervention should be performed at which part of the biological network corresponding to the pre-disease state. In other words, when a system (biological network) transitions from a stable state (healthy state) to a pre-critical state (pre-disease state), it is necessary to estimate which factor (e.g., mRNA) governing the state of the system (biological network) should be intervened in in order to restore the system (biological network) to its original stable state (healthy state). However, basic research into methods for estimating such interventions has only just begun, and many issues remain unresolved.

[0009] The present invention has been made in view of these circumstances, and its purpose is to estimate which factor, among the factors that govern the state of a system, should be intervened in order to return the system to its original stable state immediately before the system is about to transition from a stable state to a critical state.

[0010] In order to solve the above problem, one aspect of the present invention provides an intervention device that suppresses fluctuations in a network in which nodes having state quantities are connected to each other, and includes a state quantity sample acquisition unit that acquires multiple samples of the state quantities, an estimation unit, an intervention unit, and a control unit. The estimation unit selects an eigenvector related to an eigenvalue of a covariance matrix of the multiple acquired samples as an estimate of a dominant right eigenvector, and the control unit controls the intervention unit to intervene in the network in a direction that suppresses the state quantity corresponding to a component of the eigenvector related to the selected eigenvalue.

[0011] The term "dominant right eigenvector" will be explained in the detailed description section.

[0012] In one embodiment, the control unit may control the intervention unit so that, when fluctuations in the network are not suppressed, the intervention unit intervenes in the network in a direction that promotes a state quantity corresponding to a component of the selected eigenvector.

[0013] In one embodiment, the estimation unit may select the eigenvector associated with the largest eigenvalue among the eigenvalues ​​of the covariance matrix of the acquired samples as an estimate of the dominant right eigenvector.

[0014] In one embodiment, the control unit may control the intervention unit to intervene in the network with respect to a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector related to the largest selected eigenvalue.

[0015] In one embodiment, the control unit may control the intervention unit so that, when network fluctuations are not suppressed, the intervention unit intervenes in the network in a direction to suppress state quantities corresponding to components different from the components of the selected eigenvector.

[0016] In one embodiment, the network may be a biological network in which genes are connected to one another.

[0017] In one embodiment, the state quantity may be the expression level of mRNA of a gene.

[0018] Another aspect of the present invention is a method for suppressing fluctuations in a network in which nodes having state quantities are connected to one another, the method including a state quantity sample acquisition step of acquiring a plurality of samples of the state quantities, an estimation step, and an intervention step. The estimation step selects an eigenvector related to the eigenvalues ​​of a covariance matrix of the acquired plurality of samples as an estimate of a dominant right eigenvector, and the intervention step performs intervention on the network in a direction to suppress the state quantity corresponding to a component of the selected eigenvector.

[0019] In one embodiment, when the fluctuations of the network are not suppressed, the intervention step may involve performing an intervention on the network in a direction that promotes a state quantity corresponding to a component of the selected eigenvector.

[0020] In one embodiment, the estimation step may select the eigenvector associated with the largest eigenvalue among the eigenvalues ​​of the covariance matrices of the acquired samples as the estimate of the dominant right eigenvector.

[0021] In one embodiment, the intervention step may involve performing an intervention on a state quantity of the network that corresponds to the component with the largest absolute value among the components of the eigenvector associated with the largest selected eigenvalue.

[0022] In one embodiment, when the fluctuations of the network are not suppressed, the intervention step may involve intervening in the network in a direction to suppress a state quantity corresponding to a component other than the component of the eigenvector associated with the largest selected eigenvalue.

[0023] Another aspect of the present invention is a computer program. The program causes a computer to execute a method including a state quantity sample acquisition step of acquiring a plurality of state quantity samples, an estimation step, and an intervention step. The estimation step selects an eigenvector related to eigenvalues ​​of a covariance matrix of the acquired plurality of samples as an estimate of a dominant right eigenvector. The intervention step performs intervention on the network in a direction that suppresses the state quantity corresponding to a component of the selected eigenvector.

[0024] In one embodiment, the program may cause a computer to execute a method in which, when fluctuations in the network are not suppressed, the intervention step involves intervening in the network in a direction that promotes a state quantity corresponding to a component of the selected eigenvector.

[0025] In one embodiment, the program may cause a computer to execute a method in which, when fluctuations in the network are not suppressed, the intervention step intervenes in the network in a direction to suppress state quantities corresponding to components different from the components of the selected eigenvector.

[0026] Any combination of the above components, and any transformation of the present invention into an apparatus, method, system, recording medium, computer program, etc., are also valid aspects of the present invention.

[0027] According to the present invention, it is possible to estimate which of the factors governing the state of a system should be intervened in order to return the system to its original stable state immediately before the system is about to transition from a stable state to a critical state.

[0028] 11(a) is a diagram showing the distribution of eigenvalues ​​of matrix A on the complex plane. FIG. 11(b) is a schematic diagram illustrating the concept of the technology according to the present disclosure. FIG. 11(c) is a schematic diagram illustrating the concept of the technology according to the present disclosure. FIG. 11(d) is a functional block diagram of an intervention device according to a first embodiment. FIG. 11(c) is a functional block diagram of intervention devices according to a second, third, and fourth embodiment. FIG. 11(d) is a flowchart showing the processing procedure of an intervention method according to a fifth embodiment. FIG. 11(d) is a flowchart showing the processing procedure of an intervention method according to a sixth embodiment. FIG. 11(d) is a flowchart showing the processing procedure of an intervention method according to a seventh embodiment. FIG. 11(e) is a diagram showing a network structure used in a simulation. FIG. 11(f) is a diagram showing the state of each node in the network. FIG. 11(a) is a diagram showing the value of each component of a dominant right eigenvector. FIG. 11(b) is a diagram showing the value of each component of a dominant left eigenvector. FIG. 11(f) is a graph showing the results of analyzing, by real-time PCR, the effects of a WRS recombinant protein and an anti-WRS antibody on DNB gene expression. Graph showing the results of analyzing the influence of a WRS recombinant protein and an anti-WRS antibody on DNB gene expression by real-time PCR. Graph showing the results of analyzing the influence of a WRS recombinant protein and an anti-WRS antibody on DNB gene expression by real-time PCR. Graph showing the results of analyzing the influence of a WRS recombinant protein and an anti-WRS antibody on DNB gene expression by real-time PCR. Graph showing the results of analyzing the influence of a WRS recombinant protein and an anti-WRS antibody on DNB gene expression by real-time PCR.

[0029] The present invention will be described below based on preferred embodiments with reference to the drawings. In the embodiments and modifications, identical or equivalent components and members are designated by the same reference numerals, and redundant description will be omitted where appropriate. The dimensions of the components in the drawings are enlarged or reduced as appropriate for ease of understanding. Some components that are not important for explaining the embodiments are omitted from the drawings. Terms including ordinal numbers such as "first" and "second" are used to describe various components, but these terms are used only to distinguish one component from another and do not limit the components.

[0030] Before describing specific embodiments, the basic findings will first be described. In the above-mentioned DNM theory, a healthy state and a diseased state are considered to be different energetically stable states. Furthermore, attention is focused on the pre-disease state, which occurs when a healthy state changes suddenly to a diseased state. In the pre-disease state, the degree of stability is weaker than in the healthy state, and fluctuations in biological data such as mRNA expression levels become greater. If this state is left unchecked, it will transition to a diseased state, which is another stable state. From an energetic perspective, it can be seen that a great deal of effort is required to restore a diseased state to a healthy state.

[0031] Therefore, the idea is to provide treatment at the pre-disease stage, reducing fluctuations to the same level as a healthy state, and thereby returning the patient to their original healthy state. DNM theory extracts multiple genes (hereinafter referred to as "DNM genes" or "DNM nodes") with high correlations between their fluctuations and the expression levels of other genes from mRNA expression data and focuses on the gene network they form. DNM intervention involves identifying dominant genes among all genes, including these DNM genes, in order to reduce the fluctuations of the DNM genes, and intervening in their expression (i.e., suppressing or promoting them). However, as will be discussed later, genes with high fluctuations are not necessarily the targets of intervention. From this perspective, the DNM intervention problem boils down to the problem of estimating "which genes" and "whether to suppress or promote the expression of the gene" based on mRNA and other expression data from the gene network.

[0032] From the perspective of control theory, the DNM intervention problem can be understood as the restabilization of biodynamics. In other words, a previously stable healthy state is considered to be in a state in which the stability of biodynamics decreases due to changes in external factors such as reaction constants in gene expression, resulting in a pre-disease state. DNM intervention is about restabilizing this once-decreased stability, i.e., raising it again. The important question here is which gene (or genes) mRNA expression should be suppressed or promoted in order to control stability.

[0033] To solve this problem, we consider the phenomenon of mRNA expression in a gene network as the interaction of genes with each other in terms of mRNA expression. Based on this, we interpret the stability as equivalent to the stability of a dynamical system (differential equation) that determines the expression rate of each mRNA.

[0034] Let's explain the above intuitively using an analogy. Imagine a situation in which multiple balls are connected by springs (asymmetric springs, to be precise) to form a network. In this case, let's assume that one ball is swaying the most, and consider which ball's fluctuations need to be suppressed to suppress the overall fluctuation. Even if the ball that is swaying the most is suppressed, it does not necessarily mean that the fluctuations of the ball next to it will be weakened. This is because if the spring constant of the ball connected to it is low or if the mass of the ball next to it is large, the effect of suppressing the fluctuation of that ball will hardly be transmitted to the neighboring ball. In this way, in the case of a network of balls connected by springs, it can be seen that which ball should be suppressed to suppress the overall fluctuation depends on the structure of the entire network.

[0035] The above analogy can also be applied to gene networks. However, it is important to note that in the case of high-dimensional and complex network structures such as gene networks, it is generally possible to grasp only a portion of the overall structure. Furthermore, it is often difficult to obtain sufficient numbers of gene mRNA expression levels over time, and only a few samples at most can be obtained at each time point.

[0036] The following description focuses on biological systems as target systems. Biological systems are modeled as networks. In this case, the nodes that are connected to each other to form the network are genes. Each gene has an mRNA expression level as a state quantity. However, the networks that are the subject of this disclosure are not limited to this, and any type of network can be used, such as crustal systems, meteorological systems, ecosystems, transportation systems, power systems, and financial systems, as long as the network is composed of nodes with state quantities.

[0037] Hereinafter, unless otherwise specified, when the number of nodes is n, z(t) is an n-dimensional column vector representing the expression level of mRNA. ｅ is an n-dimensional column vector that represents the stable point (equilibrium point) of the mRNA expression level. x(t) = z(t) - z ｅis an n-dimensional column vector that represents the deviation (fluctuation) of the mRNA expression level from the stable point (equilibrium point). Ｔ Let (t) denote the column vector obtained by transposing x(t). Also, x(t) may be abbreviated simply as x.

[0038] Generally, a high-dimensional network system can be modeled as a nonlinear dynamical system, but in this specification, the time change of x(t) can be approximated by the following linear system: A is a system matrix that represents the interaction between state quantities. Specifically, the component a of A ｉｊ represents the influence of the jth state quantity on the ith state quantity. In general, the influence of the ith state quantity on the jth state quantity is different from the influence of the jth state quantity on the ith state quantity, so A is an asymmetric matrix. Also, A is a stable matrix, and the real parts of all eigenvalues ​​are assumed to be negative. ω(t) represents noise. The dot x(t) on the left side represents the time derivative of x(t). In this specification, the intervention of mRNA is assumed to be intervention in the diagonal elements of the system matrix A. This is because the components a of matrix A ｉｉ This corresponds to changing the value of

[0039] As an example, consider the case where A is: When this system reaches a pre-critical state (i.e., when the fluctuation of the entire system is large), we consider intervening in the system to suppress the fluctuation and return it to its original stable state. For example, if we know that the factor that contributes most to the fluctuation of the entire system is "the effect of the first state variable on the first state variable (itself)," then we can add It is believed that the fluctuation of the entire system can be suppressed by adding an intervention expressed as follows. In other words, the (1, 1) element a of matrix A １１ (The influence of the first state quantity on the first state quantity) As a result, the system matrix A is Then, the real part of the eigenvalue with the largest real part becomes smaller, and the system returns to a stable state.

[0040] The above example assumes that it is known which interaction between state variables contributes most to the overall system fluctuation. However, as mentioned above, high-dimensional network structures such as gene networks can only be partially understood, and only a limited number of state variables can be sampled. Therefore, it is unclear which interaction between state variables contributes most to the overall system fluctuation. In other words, it is difficult to identify the system matrix A. Therefore, it is impossible to obtain an exact solution by solving the stabilization problem used in conventional control theory. This has been a major factor that has traditionally made it difficult to restore a pre-critical state to a stable state through intervention. To address this unresolved issue, the inventors discovered that by following the procedure below, it is possible to estimate the state variables that contribute most to the fluctuation, and therefore to stabilize the system by intervening in these state variables.

[0041] (Step 1) Obtain samples of expression levels z(t) of multiple mRNAs in a pre-disease state.

[0042] The number of samples may be relatively small. For example, in this specification, several to several tens of samples are assumed, but the number of samples is not limited to this and may be any number. As described above, z ｅ When (t) is an n-dimensional column vector representing the stable point (equilibrium point) of mRNA expression level, x(t) = z(t) - z ｅ Let (t) be an n-dimensional column vector representing the deviation (fluctuation) of the mRNA expression level from a stable point (equilibrium point).

[0043] (Step 2) Covariance matrix E[x(t)x of z(t) Ｔ (t)] and E[x(t)x Ｔ (t)], the largest eigenvalue λ ｄ eigenvector v ｄ is calculated and selected as the estimate of the dominant right eigenvector. １ (M) represents the eigenvalue with the largest real part of the matrix M, and the eigenvalue λ of the system matrix A １ The right eigenvector corresponding to (A) is called the dominant right eigenvector, and v １The left eigenvector is called the dominant left eigenvector, and w １ It is written as follows.

[0044] λ ｄ is sometimes called the "dominant eigenvalue". Ｔ (t)] is a covariance matrix for a limited number of mRNA expression samples, and is therefore sometimes called a "sample covariance matrix." ｄ is sometimes called the "dominant eigenvector of the sample covariance matrix." Note that since the sample covariance matrix is ​​a symmetric matrix, the right eigenvector and the left eigenvector are the same. In the pre-critical state (pre-disease state), this v ｄ is v １ In other words, the dominant eigenvector v of the sample covariance matrix is ​​known to be a good approximation of ｄ is the dominant right eigenvector v of the system matrix A. １ This can be considered an estimate of the

[0045] In contrast, the dominant left eigenvector w １ So far, no effective method for estimating this has been found.

[0046] (Step 3) The dominant eigenvector v of the sample covariance matrix obtained in Step 2 ｄ The component with the largest absolute value is v ｄ (i ｍａｘ ) and this i ｍａｘ The intervention is to suppress the expression of mRNA of the corresponding gene.

[0047] At this time The eigenvalues ​​of Here, the dominant right eigenvector v of matrix A is １ is the dominant eigenvector v of the sample covariance matrix ｄ It can be estimated that is negative if it is an inhibitory effect, and positive if it is a facilitative effect. On the other hand, at this stage, the dominant left eigenvector w of matrix A is calculated from the small sample data. １ There is no known method to estimate iｍａｘ It is unclear whether intervening in the corresponding genes will be highly effective, and it is also unclear in which direction (inhibitory or stimulatory) intervention should be carried out.

[0048] In the following, we will first provide a preliminary explanation, assuming that matrix A is known. Figure 1 shows the distribution of eigenvalues ​​of matrix A on the complex plane. The points marked with x are eigenvalues. When the rank of matrix A (which is assumed to be diagonalizable in this specification) is n, there are n non-zero eigenvalues, including multiplicities. Among these eigenvalues, the λ closest to the imaginary axis is １ is the dominant eigenvalue (dominant right eigenvector v １ Here, this is referred to as the eigenvalue corresponding to λ (the eigenvalue corresponding to λ ) because, when matrix A is subjected to eigenvalue decomposition, the λ closest to the imaginary axis (i.e., closest to zero) is １ This is because the component of the state related to λ is the slowest to converge, and therefore contributes most to the fluctuations. １ The closer to the imaginary axis (i.e., λ １ The closer λ is to zero, the greater the fluctuations and the closer the system approaches bifurcation. Therefore, as shown by the arrows in Figure 1, we can intervene in the system to change λ １ is moved away from the imaginary axis (i.e., λ １ It is believed that by making λ smaller, the system can be brought back to a stable state. １ λ ｓ This shows how the image has been changed to

[0049] The example in Figure 1 shows an example where an eigenvalue crosses the imaginary axis of the complex plane at a single point (the origin), such as a saddle-node bifurcation (tangent bifurcation). Therefore, in this example, the dominant eigenvalue is a real number. However, this is not limited to this case; the above concept can also be applied to cases where two complex conjugate eigenvalues ​​cross the imaginary axis, such as a Hopf bifurcation. In this case, two eigenvectors are found from the sample covariance matrix. Furthermore, genes corresponding to components with large absolute values ​​for the real and imaginary parts of the complex eigenvectors corresponding to the complex eigenvalues ​​of matrix A are candidates for intervention. In other words, two genes are candidates: one for a gene corresponding to a component with a large absolute value of the real part, and the other for a gene corresponding to a component with a large absolute value of the imaginary part. Furthermore, there are two types of candidates, inhibition and promotion, for a total of four candidates.

[0050] From the preliminary explanation, we return to the explanation of (Step 3). As mentioned above, in an actual gene network structure, matrix A is unknown, and it is not known which gene should actually be intervened in, nor is it known in which direction (repression or promotion) the intervention should be made with respect to the expression of the mRNA of that gene.

[0051] In response to this, the inventors have conducted research and have found the following. In v １ (i ｍａｘ ) is v ｄ It is estimated from w １ (i ｍａｘ )and Consider the case where is unknown. (However, The sign of is known, since it is negative for suppression and positive for promotion.) In this case, it is not possible to strictly identify effective genes. However, １ (i ｍａｘ ) has a large value, it is highly likely that fluctuations in the entire system can be suppressed by intervening in the expression of the corresponding mRNA. This is a completely new finding not found in conventional technology.

[0052] The reason why components of the dominant right eigenvector with larger absolute values ​​are more advantageous for stabilizing the network can be intuitively explained as follows: The behavior of this system is determined by the overlap of multiple modes (which can be thought of as predetermined frequency signals) that determine the rate of change. The most dominant mode that determines fluctuations in mRNA expression levels in a pre-disease state is characterized by the dominant right eigenvector. Therefore, it is thought that genes that are advantageous for intervention are among the genes corresponding to components of this eigenvector with larger absolute values.

[0053] As explained above, by following steps (Step 1) to (Step 3), the system can be returned to its original stable state with a relatively high degree of accuracy, thereby solving the problem of the present disclosure.

[0054] However, it cannot be estimated １ (i ｍａｘ If the sign of λ is reversed, intervention in the direction of suppression is performed in step 3. １ approaches the imaginary axis (i.e., λ １ In such a case, the steps (Step 1) to (Step 3) alone cannot suppress the fluctuations of the entire system, and may even increase the fluctuations. In response to this, the inventors have discovered that by performing the next step (Step 4) after (Step 3), the fluctuations can be suppressed with a higher degree of accuracy.

[0055] (Step 4) If the fluctuations cannot be suppressed in (Step 1) to (Step 3), ｍａｘ The gene corresponding to the target gene is subjected to an intervention to promote the expression of its mRNA.

[0056] In this way, by intervening in the opposite direction to that of (Procedure 3) (i.e., in the direction of promoting mRNA expression) in (Procedure 4), fluctuations can be suppressed even in cases where overall fluctuations could not be suppressed by following the procedures from (Procedure 1) to (Procedure 3).

[0057] However, it cannot be estimated １ (i ｍａｘ ) is extremely close to zero, Since the term itself becomes extremely small, this term does not contribute significantly to fluctuations. In such cases, the fluctuations of the entire system cannot be suppressed by the steps (Step 1) to (Step 4) alone. In contrast, the inventors have discovered that fluctuations can be suppressed with a higher degree of accuracy by performing the following (Step 5) after (Step 4).

[0058] (Step 5) The dominant right eigenvector v of the sample covariance matrix obtained in Step 3 ｄ The component with the second largest absolute value is v ｄ (i ｍａｘ２ ) and this i ｍａｘ２ The intervention is to suppress the expression of mRNA of the corresponding gene.

[0059] In this way (step 5), i ｍａｘ instead of i ｍａｘ２ By intervening to suppress the expression of mRNA for the gene corresponding to the above, it is possible to suppress the fluctuation even if the overall fluctuation could not be suppressed by following steps (Step 1) to (Step 4).

[0060] If the fluctuations of the entire system cannot be suppressed by steps (1) to (5), the process returns to step (4) and the same process can be repeated until the fluctuations can be suppressed.

[0061] 2 and 3 are schematic diagrams illustrating the concept of the technology according to the present disclosure described above.

[0062] 4 shows a functional block diagram of an intervention device 1 according to a first embodiment. The intervention device 1 intervenes in a network in which nodes having state quantities are connected to each other, thereby suppressing fluctuations in the network. The intervention device 1 includes a state quantity sample acquisition unit 11, an estimation unit 12, and an intervention unit 13.

[0063] The state quantity sample acquisition unit 11 acquires a plurality of state quantity samples. The number of samples may be extremely small. For example, when the target network is a gene network, the samples may be mRNA expression levels acquired by a DNA sequencer.

[0064] The estimation unit 12 calculates a covariance matrix of the multiple samples acquired by the state quantity sample acquisition unit 11. Then, the estimation unit 12 calculates an eigenvector associated with the largest eigenvalue of the covariance matrix, and selects this as an estimate of the dominant right eigenvector.

[0065] The intervention unit 13 intervenes in the target network to suppress fluctuations in the network. Specifically, the intervention unit 13 targets a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected by the estimation unit 12, and intervenes in a direction to suppress this state quantity.

[0066] According to this embodiment, when the target network transitions from a stable state to a pre-critical state (i.e., when fluctuations occur in the network), it is possible to estimate which state quantities constituting the network should be intervened in to suppress the fluctuations in order to return the network to its original stable state.

[0067] [Second embodiment] Fig. 5 shows a functional block diagram of an intervention device 2 according to a second embodiment. The intervention device 2 includes a state quantity sample acquisition unit 11, an estimation unit 12, an intervention unit 13, and a control unit 14. That is, the intervention device 2 includes the control unit 14 in addition to the configuration of the intervention device 1 shown in Fig. 4. The other configurations and operations of the intervention device 2 are common to those of the intervention device 1.

[0068] When the network fluctuation is not suppressed, the control unit 14 controls the intervention unit 13. Specifically, the control unit 14 controls the intervention unit 13 to intervene in the target network in a direction that promotes a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected by the estimation unit 12.

[0069] According to this embodiment, fluctuations in the target network can be suppressed with a higher degree of accuracy.

[0070] [Third embodiment] A third embodiment will be described with reference to Fig. 5. In the third embodiment, when fluctuations in the network are not suppressed, the control unit 14 controls the intervention unit 13 so that the intervention unit 13 performs intervention in the target network in a direction to suppress a state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected by the estimation unit 12.

[0071] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0072] [Fourth embodiment] A fourth embodiment will be described with reference to Fig. 5. In the fourth embodiment, when the fluctuation of the network is not suppressed even after intervention in a direction to suppress the state quantity corresponding to the component with the second largest absolute value, the control unit 14 controls the intervention unit 13 to intervene in the network in a direction to promote the state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected by the estimating unit 12. When the fluctuation of the network is still not suppressed, the control unit 14 controls the estimating unit 12 and the intervention unit 13 so that the intervention unit 13 intervenes in the network in a direction to suppress the state quantity corresponding to the component with the third largest absolute value among the components of the eigenvector selected by the estimating unit 12.

[0073] Similarly, if the network fluctuation is not suppressed even after intervention in the suppressing direction, the control unit 14 controls the intervention unit 13 to intervene in the promoting direction. If the network fluctuation is still not suppressed, the control unit 14 controls the estimation unit 12 and the intervention unit 13 to intervene in the suppressing direction to suppress the state quantity corresponding to the component with the next largest absolute value among the components of the eigenvector selected by the estimation unit 12. The control unit 14 repeats the above processing procedure, controlling the estimation unit 12 and the intervention unit 13, until the network fluctuation is suppressed.

[0074] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0075] 6 is a flowchart showing the processing steps of an intervention method according to a fifth embodiment. This method includes step S1 of acquiring samples of state quantities, step S2 of estimating a dominant right eigenvector, and step S3 of implementing suppression intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2.

[0076] In step S1, the method takes a number of samples of the state quantities, which may be very small.

[0077] In step S2, the method calculates a covariance matrix of the samples acquired in step S1, calculates the eigenvector associated with the largest eigenvalue of the covariance matrix, and selects this as an estimate of the dominant right eigenvector.

[0078] In step S3, this method intervenes in the target network to suppress fluctuations in the network. Specifically, in step S3, the state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2 is targeted, and intervention is performed to suppress this state quantity.

[0079] According to this embodiment, when the target network transitions from a stable state to a pre-critical state (i.e., when fluctuations occur in the network), it is possible to estimate which state quantities constituting the network should be intervened in to suppress the fluctuations in order to return the network to its original stable state.

[0080] 7 is a flowchart showing the processing steps of an intervention method according to a sixth embodiment. This method includes step S1 of acquiring samples of a state quantity, step S2 of estimating a dominant right eigenvector, step S3 of implementing a suppression intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, step S4 of determining whether fluctuations have been suppressed, and step S5 of implementing a promotion intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2. That is, the intervention method according to the sixth embodiment includes step S4 and step S5 following step S3 of the intervention method according to the fifth embodiment.

[0081] In step S4, the method determines whether fluctuations in the target network have been suppressed by the processes in steps S1 to S3. If the determination result is affirmative, the process ends. If the determination result is negative, the process proceeds to step S5.

[0082] In step S5, the method targets the state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, and performs intervention in a direction to promote this.

[0083] According to this embodiment, fluctuations in the target network can be suppressed with a higher degree of accuracy.

[0084] 8 is a flowchart showing the processing steps of an intervention method according to a seventh embodiment. This method includes step S1 of acquiring samples of state quantities, step S2 of estimating a dominant right eigenvector, step S3 of implementing a suppression intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, step S4 of determining whether fluctuations have been suppressed, step S5 of implementing a promotion intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, step S6 of determining whether fluctuations have been suppressed, and step S7 of implementing a suppression intervention on a state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected in step S2. That is, the intervention method according to the seventh embodiment includes step S6 and step S7 following step S5 of the intervention method according to the sixth embodiment.

[0085] In step S6, the method determines whether fluctuations in the target network have been suppressed by the processes in steps S1 to S5. If the determination result is affirmative, the process ends. If the determination result is negative, the process proceeds to step S7.

[0086] In step S7, the method targets the state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected in step S2, and intervenes to suppress this.

[0087] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0088] [Eighth embodiment] In the eighth embodiment, if the fluctuation of the network is not suppressed even after intervention in a direction to suppress a state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected in step S2, an intervention step S5 intervenes in the network in a direction to promote a state quantity corresponding to the second largest component among the components of the eigenvector selected in estimation step S2. If the fluctuation of the network is still not suppressed, an intervention step S5 intervenes in the network in a direction to suppress a state quantity corresponding to the component with the third largest absolute value among the components of the eigenvector selected in estimation step S2.

[0089] Similarly, if the network fluctuation is not suppressed even after intervention in the suppression direction, intervention in the promotion direction is performed in the intervention step S5. If the network fluctuation is still not suppressed, intervention step S5 intervenes in the direction of suppressing the state quantity corresponding to the component with the next largest absolute value among the components of the eigenvector selected in step S2. Intervention step S5 repeats the above processing procedure until the network fluctuation is suppressed.

[0090] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0091] 6, the ninth embodiment is a computer program that causes a computer to execute a method including step S1 of acquiring samples of a state quantity, step S2 of estimating a dominant right eigenvector, and step S3 of implementing suppression intervention on a state quantity corresponding to a component with the largest absolute value among the components of the eigenvector selected in step S2.

[0092] According to this embodiment, when the target network transitions from a stable state to a pre-critical state (i.e., when fluctuations occur in the network), in order to return the network to its original stable state, a method can be implemented as a software program to estimate which state quantities constituting the network should be intervened in to suppress the fluctuations.

[0093] 7 , the program causes a computer to execute a method including the steps of: acquiring a sample of a state quantity, estimating a dominant right eigenvector, estimating a dominant right eigenvector, implementing a suppression intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, determining whether fluctuations have been suppressed, and implementing a promotion intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2.

[0094] According to this embodiment, a method that can suppress fluctuations in a target network with higher accuracy can be implemented as a software program.

[0095] 8 , the program causes a computer to execute a method including the steps of: acquiring a sample of a state quantity, estimating a dominant right eigenvector, estimating a dominant right eigenvector, suppressing an intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2, determining whether fluctuations have been suppressed; promoting an intervention on a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector selected in step S2; determining whether fluctuations have been suppressed; and suppressing an intervention on a state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected in step S2.

[0096] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0097] [Twelfth Embodiment] The twelfth embodiment is also a computer program. In this program, if the fluctuation of the network is not suppressed even after intervention in a direction to suppress a state quantity corresponding to the component with the second largest absolute value among the components of the eigenvector selected in step S2, an intervention step S5 intervenes in the network in a direction to promote the state quantity corresponding to the second largest component among the components of the eigenvector selected in step S2. If the fluctuation of the network is still not suppressed, an intervention step S5 intervenes in the network in a direction to suppress a state quantity corresponding to the component with the third largest absolute value among the components of the eigenvector selected in step S2.

[0098] Similarly, if the network fluctuation is not suppressed even after intervention in the suppression direction, intervention in the promotion direction is performed in the intervention step S5. If the network fluctuation is still not suppressed, intervention step S5 intervenes in the direction of suppressing the state quantity corresponding to the component with the next largest absolute value among the components of the eigenvector selected in step S2. Intervention step S5 repeats the above processing procedure until the network fluctuation is suppressed.

[0099] According to this embodiment, fluctuations in the target network can be suppressed with even higher accuracy.

[0100] [Verification Experiments] The inventors conducted experiments to verify the effectiveness of the technology disclosed herein. (Experiment 1) Experiment 1 is a numerical simulation. Figure 9 shows the network structure used in the simulation. This network was generated using the Holme-Kim model and has scale-free properties and high clustering properties. The number of nodes is 20. Each node has a self-loop with a probability of 0.1.

[0101] Figure 10 shows the state of each node in the network. The horizontal axis represents the calculation steps in 10 steps. ５The bifurcation parameter γ is expressed in units of 1.329. The patient is in a healthy state until it reaches 1.329, but when it changes to 1.33, it suddenly becomes a diseased state. In other words, a bifurcation phenomenon occurs here. A sample of the state quantity is obtained from the steady state just before the bifurcation. Here, the number of samples is set to 5.

[0102] FIG. 11(a) is a diagram showing the values ​​of each component of the dominant right eigenvector. This diagram shows results based on a sample covariance matrix of five samples, results based on the covariance matrix for all samples (20 samples), and true values. The results based on the covariance matrix for all samples (20 samples) and the true values ​​are in excellent agreement. Although there are some discrepancies, the results based on a sample covariance matrix of five samples and the true values ​​are generally in good agreement. This shows that the technique of the present disclosure can correctly estimate the dominant right eigenvector. Both results show that the contribution of the sixth gene is dominant.

[0103] 11(b) is a diagram showing the values ​​of each component of the dominant left eigenvector. As mentioned above, the dominant left eigenvector cannot be estimated, so only the true value is shown. As shown in the diagram, it can be seen that the contribution of the 10th gene is dominant.

[0104] 12 is a diagram showing the product of the dominant right eigenvector and the dominant left eigenvector. From this, it is considered that the contributions of the sixth and tenth genes are ultimately dominant, and that network fluctuations can be suppressed by suppressing these genes.

[0105] (Experiment 2) In Experiment 2, colitis was induced in male BALB / c mice by allowing them to freely drink a 3% aqueous solution of dextran sulfate sodium (DSS) (MW: 36-50 kDa) (manufactured by MP Biomedicals) for 7 days. Gene expression profiles of mouse colon tissues 0, 1, 3, 5, and 7 days after the start of DSS drinking were comprehensively analyzed by microarray analysis as described below, and these data were analyzed using dynamic network biomarker (DNB) theory (DNB analysis).

[0106] Microarray analysis was performed using a SurePrint G3 Mouse Gene Expression 8x60K Microarray Kit (Agilent Technologies). Colon tissues were isolated from mice 0, 1, 3, 5, and 7 days after the start of DSS drinking water (n = 5 for each time point), and total RNA extracted from the colon tissue was used for microarray analysis. Microarray analysis was performed according to the kit's protocol.

[0107] In the DNB analysis in this experiment, first, genes whose expression levels were significantly changed were selected from all genes using the gene expression data in mouse colon tissue obtained by the microarray analysis. Next, the correlation coefficients between the expression levels of the selected genes were calculated. Next, genes with high correlations among the selected genes were clustered. As a result, 27 DNB genes were detected: Mcpt1, Mcpt4, Ifit2, Cd274, Trim30d, Tnfsf10, Ccn1, Wars, Tap2, Plppr3, Ube2l6, Bst2, Cxcl16, B2m, 4933412E12Rik, H2-T23, Znfx1, H2-M2, H2-Bl, H2-Q2, Gm11127, H2-K2, H2-Q8, H2-Q7, H2-Q6, H2-D1, and Fbxo6.

[0108] The 27 genes detected by the DNB analysis were scored using the method disclosed herein. Specifically, from the expression data of all target genes, only the expression data of the DNB gene on day 3 was extracted, and the eigenvector for the largest eigenvalue of the covariance matrix of that data was calculated. The components of this eigenvector were sorted in descending order of absolute value. After dividing each value by the maximum value, the proportion of variation related to the largest eigenvalue that explains the variance of the entire data was calculated and multiplied. The results are shown in Table 1.

[0109] Table 1

[0110] Of the genes shown in Table 1, Mcpt1 and Mcpt4 are not present in humans, Ifit2, Cd274, and Tnfsf10 have been detected in conventional analyses using the average gene expression level as an index, and Trim30d, Ccn1, Plppr3, 4933412E12Rik, H2-T23, H2-M2, H2-Bl, H2-Q2, Gm11127, H2-K2, H2-Q8, H2-Q7, H2-Q6, and H2-D1 are not present in the above IBD data. Therefore, the inventors focused on Wars among the genes shown in Table 1 and examined whether it is a gene of pathophysiological significance in IBD below. Wars is also present in humans, and was not detected in conventional analyses using the average gene expression level as an index, but was first detected by DNB analysis, and was a highly ranked gene.

[0111] (Experiment 3) Experiment 3 was an experiment to analyze the influence of WRS recombinant protein and anti-WRS antibody on DNB gene expression. The influence of WRS recombinant protein and anti-WRS antibody on DNB gene expression was analyzed by real-time PCR according to the following procedure.

[0112] DSS-induced colitis model mice administered WRS recombinant protein or anti-WRS antibody were prepared as follows. WRS recombinant protein (rWRS; manufactured by CLOUD-CLONE) was dissolved in saline and administered intraperitoneally at 20 μg per mouse once daily on days 0, 2, 4, and 6 after the start of DSS drinking. As a control, vehicle (saline) was administered in the same manner. Anti-WRS antibody (Anti-WRS Ab) was prepared using Cosmo Bio's first antibody and administered intraperitoneally at 150 μg per mouse once daily on days 0 and 2 after the start of DSS drinking. As a control, a control IgG antibody (rabbit IgG control (R&D Systems; catalog number: AB-105-C)) was administered in the same manner. Three days after the start of DSS drinking, real-time PCR was performed to analyze the expression levels of B2m, Bst2, Ccn1, Cd274, Cxcl16, Fbxo6, H2-D1, H2-K2, H2-M2, H2-Q2, H2-Q6, H2-Q7, H2-T23, Ifit2, Mcpt1, Mcpt4, Tap2, Trim30d, Ube2l6, Wars, and Znfx1.

[0113] For real-time PCR, RNA was extracted from the colon tissue using Sepasol RNA I Super G (Nacalai Tesque) according to the manufacturer's protocol, and the RNA concentration and quality were measured using a Nano Drop (ND-1000; Thermo Fisher Scientific). Reverse transcription was performed to synthesize cDNA from the extracted RNA using the PrimeScript RT reagent Kit (Takara Bio).

[0114] Using the synthesized cDNA as a template, the expression level of the target gene was measured using TB Green Premix Ex Tag (manufactured by Takara Bio Inc.). The expression level of each target gene was normalized with the expression level of the TBP (TATA binding protein) gene, which is an internal standard. －ΔΔＣｔThe calculation was performed by the method. The primer combinations for each target gene listed in Table 2 were used as primers to amplify the target gene and TBP gene. Primers for each target gene were purchased from Takara Bio Inc. As a control, normal BALB / c mice were also used to analyze the expression of the genes listed in Table 2 below by real-time PCR. The results are shown in Figures 13 to 18.

[0115] Table 2

[0116] As shown in Figures 13 to 18, administration of WRS recombinant protein to colitis model mice suppressed fluctuations in the expression of Wars, as well as B2m, Cxcl16, and Ube2l6 in the pre-disease state (3 days after the start of DSS drinking).

[0117] The present invention has been described above based on the embodiments. These embodiments are merely examples, and it will be understood by those skilled in the art that various modifications are possible in the combination of each component and each treatment process, and that such modifications are also within the scope of the present invention.

[0118] When understanding the abstract technical ideas of the embodiments and modifications, the technical ideas should not be interpreted as being limited to the contents of the embodiments and modifications. The above-described embodiments and modifications are merely illustrative examples, and many design modifications, such as changes, additions, and deletions of components, are possible. In the embodiments, the contents in which such design modifications are possible are emphasized by adding the notation "embodiment." However, design modifications are also permitted even in contents without such notation.

[0119] The technology disclosed herein can be widely used for early response to changes in natural and social phenomena, such as early treatment of diseases, early response to natural disasters and environmental changes such as earthquakes and climate change, early response to traffic congestion, early response to fluctuations in electricity demand, and early response to fluctuations in exchange rates and stock prices.

[0120] 1. Intervention device, 2. Intervention device, 11. State sample acquisition unit, 12. Estimation unit, 13. Intervention unit, 14. Control unit, S1. Step of acquiring a sample of a state quantity, S2. Step of estimating a dominant right eigenvector, S3. Step of implementing an intervention in a suppressive direction on a state quantity corresponding to the component with the largest absolute value, S4. Step of determining whether fluctuations have been suppressed, S5. Step of implementing an intervention in a promoting direction on a state quantity corresponding to the component with the largest absolute value, S6. Step of determining whether fluctuations have been suppressed, S7. Step of implementing an intervention in a suppressive direction on a state quantity corresponding to the component with the second largest absolute value.

Claims

1. An intervention device for suppressing fluctuations in a network in which nodes having state quantities are connected to each other, comprising: a state quantity sample acquisition unit that acquires multiple samples of the state quantities; an estimation unit; an intervention unit; and a control unit, wherein the estimation unit selects an eigenvector related to an eigenvalue of a covariance matrix of the multiple acquired samples as an estimate of a dominant right eigenvector, and a pre-control unit controls the intervention unit to intervene in the network in a direction that suppresses the state quantity corresponding to a component of the eigenvector related to the selected eigenvalue.

2. The intervention device described in claim 1, characterized in that the control unit controls the intervention unit so that, when fluctuations in the network are not suppressed, the intervention unit intervenes in the network in a direction that promotes a state quantity corresponding to a component of the selected eigenvector.

3. The intervention device according to claim 1 or 2, characterized in that the estimation unit selects the eigenvector associated with the largest eigenvalue among the eigenvalues ​​of the covariance matrix of the acquired multiple samples as an estimate of the dominant right eigenvector.

4. The intervention device according to claim 3, characterized in that the control unit controls the intervention unit to intervene in the network with respect to a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector related to the selected largest eigenvalue.

5. The intervention device described in claim 1 or 2, characterized in that when the fluctuations of the network are not suppressed, the control unit controls the intervention unit so that the intervention unit intervenes in the network in a direction that suppresses a state quantity corresponding to a component different from the component of the selected eigenvector.

6. The intervention device according to claim 1, wherein the network is a biological network in which genes are connected to each other.

7. The intervention device according to claim 6, wherein the state quantity is the expression level of mRNA of the gene.

8. An intervention method for suppressing fluctuations in a network in which nodes having state quantities are connected to each other, comprising: a state quantity sample acquisition step for acquiring a plurality of samples of the state quantities; an estimation step; and an intervention step, wherein the estimation step selects an eigenvector related to the eigenvalues ​​of a covariance matrix of the acquired plurality of samples as an estimate of a dominant right eigenvector, and the intervention step performs intervention on the network in a direction to suppress the state quantities corresponding to the components of the selected eigenvector.

9. The method according to claim 8, characterized in that when the fluctuations of the network are not suppressed, the intervention step involves intervening in the network in a direction that promotes the state quantity corresponding to the component of the selected eigenvector.

10. The method according to claim 8 or 9, characterized in that the estimation step selects the eigenvector associated with the largest eigenvalue among the eigenvalues ​​of the covariance matrix of the acquired plurality of samples as an estimate of the dominant right eigenvector.

11. The method according to claim 10, wherein the intervention step involves intervening in the network with respect to a state quantity corresponding to the component with the largest absolute value among the components of the eigenvector related to the selected largest eigenvalue.

12. The method of claim 10, characterized in that when the fluctuations of the network are not suppressed, the intervention step performs an intervention on the network in a direction to suppress a state quantity corresponding to a component other than the component of the eigenvector related to the selected largest eigenvalue.

13. A program for suppressing fluctuations in a network in which nodes having state quantities are connected to each other, comprising: a state quantity sample acquisition step for acquiring a plurality of samples of the state quantities; an estimation step; and an intervention step, wherein the estimation step selects an eigenvector related to the eigenvalues ​​of a covariance matrix of the acquired plurality of samples as an estimate of a dominant right eigenvector; and the intervention step causes a computer to execute a method for intervening in the network in a direction that suppresses the state quantities corresponding to the components of the selected eigenvector.

14. The program described in claim 13, characterized in that when the fluctuations of the network are not suppressed, the intervention step causes the computer to execute a method of intervening in the network in a direction that promotes a state quantity corresponding to the component of the selected eigenvector.

15. The program described in claim 13, characterized in that when the fluctuations of the network are not suppressed, the intervention step causes the computer to execute a method of intervening in the network in a direction that suppresses state quantities corresponding to components different from the components of the selected eigenvector.

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