Method for creating a network model of a dynamic system of interdependent variables from system observations

• US 6,480,814 B1
• Filed: 04/29/1999
• Issued: 11/12/2002
• Est. Priority Date: 10/26/1998
• Status: Expired due to Fees

• Current Assignee: International Business Machines Corporation
• Original Assignee: NuTech Solutions, Inc. (International Business Machines Corporation)
• Inventors: Levitan, Bennett Simeon
• Primary Examiner(s): Teska, Kevin J.
• Assistant Examiner(s): Broda, Samuel
• Application Number:
  US09/301,615
• Time in Patent Office: 1,293 Days
• Field of Search: 703/3, 703/11, 703/6, 703/19, 703/21, 703/1-2, 706/60, 706/34, 359/139, 379/112, 382/157
• US Class Current: 703/2
• CPC Class Codes: G06N 3/126 Evolutionary algorithms, e.g. genetic algorithms or genetic programming

Abstract

The present invention provides a method to create a Boolean or multilevel logic network model of a dynamic system of interdependent variables from observed system states transitions that can

(1) operate with data consisting of only a fraction of all possible transitions,

(2) accommodate measurement error on these transitions

(3) produce a probability distribution over network functions (rather than simply giving one set of network functions that match the data)

(4) support asynchronous activation times for different genes

(5) support varying delays between gene activation and gene expression for different genes

(6) accommodate the stochastic nature of gene network operations

(7) support varying degrees of gene activation (not simply Boolean activation states) and

(8) incorporate prior knowledge of the nature and limitations of the actual network functions being modeled.

Claims

1. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising the steps of:

defining N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

defining N network rules ƒ

_i, i=0, . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

defining N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

defining at least one likelihood function expressing the probability of making the system observations for said possible network rules;

determining N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system; and

defining at least one placeholder value u to represent an unknown expression level for said N network variables, x_i, i=0, . . . N−

1.

2. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 1 wherein said N prior distributions are uniform probability distributions.

3. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising the steps of:

defining N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

defining N network rules ƒ

_i, i=0 . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

defining N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

defining at least one likelihood function expressing the probability of making the system observations for said possible network rules;

determining N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system; and

selecting one or more of said posterior distributions having the greatest probability to identify said network model of the real world system;

wherein said defining N prior distributions step comprises the step of defining said N prior probability distributions as an expression of a plurality of abstract properties of said N network rules.

4. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 3 wherein the network model comprises at least one of a multi-level logic network model, a boolean network model or a genetic regulatory network.

5. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 3 wherein said defining said N prior probability distributions as an expression of a plurality of abstract properties step further comprises the steps of:

defining N influence variables K_i0≦

K_i≦

N corresponding to said N network rules ƒ

_i, each of said influence variables is the number of said N network variables effecting said corresponding network rule ƒ

_i;

defining N canalizing variables C_i, 0≦

C_i≦

N;

corresponding to said N network rules, each of said canalizing variables is the number of said N network variables for which one value completely determines the output of said corresponding network rule ƒ

_i; and

defining N equivalence variables P_i, 0≦

K_i≦

N corresponding to said N network rules, each of said equivalence variables is the fraction of the outputs of said corresponding network rule ƒ

_i having the same value.

6. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world as in claim 5 wherein said defining said N prior probability distributions as an expression of a plurality of abstract properties step further comprises the step of decomposing said N prior probability distributions as:

7. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world as in claim 5 wherein said defining said N prior probability distributions as an expression of a plurality of abstract properties step further comprises the step of decomposing said N prior probability distributions as:

8. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world as in claim 5 wherein said defining said N prior probability distributions as an expression of a plurality of abstract properties step further comprises the step of decomposing said N prior probability distributions as:

9. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 5 further comprising the step of defining said space of possible network rules.

10. A method for creating a network model of a real world system of interacting components having a plurality of observations of the real world system as in claim 9 wherein said defining said space of possible network rules step comprises the steps of:

identifying at least one constraint on said space of possible network rules;

checking said space of possible network rules against said at least one constraint to identify a set of unrealistic network rules; and

removing said set of unrealistic rules from consideration for faster identification of said network model of the real world system.

11. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 10 wherein said at least one constraint comprises a set of limits on said plurality of abstract properties.

12. A method for creating a network mode of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 11 wherein said at least one constraint comprises a maximum value for said influence variable K_i.

13. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 10 wherein said at least one constraint comprises a set of dependency requirements among said N network variables, x_i, i=0, . . . N−

1.

14. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising the steps of:

defining N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

defining N network rules ƒ

_i, i=0 . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

defining N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

defining at least one likelihood function expressing the probability of making the system observations for said possible network rules;

determining N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system;

defining a sequence of discrete time steps t, t−

1, . . . t−

T individually referenced by τ

; and

defining said network state {right arrow over (x)}^τ

of said network model at said discrete time step τ

as a vector of said values, x_i^τ

ε

{v_i, . . . v_m}, i=0, . . . N−

1 at said time step τ

.

15. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 14 wherein said N network rules are defined as x_i^t=ƒ

_i({right arrow over (x)}<sup>t−

1</sup>, {right arrow over (x)}<sup>t−

2</sup>, {right arrow over (x)}<sup>t−

3</sup>, . . . {right arrow over (x)}<sup>t−

T</sup>) wherein each of said N network rules ƒ

_i, i=0 . . . N−

1 determines the value for said corresponding variable x_i at said time step t, x_i^t, from said network state {right arrow over (x)}^τ

said time steps, τ

=t=1 . . . t−

T.

16. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 15 wherein the observations are noiseless and said at least one likelihood function is defined as:

$\begin{array}{c}p<br><br>\left(T<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>p<br><br>\left({\stackrel{\to <br><br>}{x}}^j\to <br><br>{\stackrel{\to <br><br>}{y}}^j<br><br>Q,\mathrm{\zeta <br><br>}\right)\\ =\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>\underset{i=0}{\overset{N-1}{\prod <br><br>}}<br><br>\mathrm{\delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^j},y_i^j\right)\end{array}$

wherein T={{right arrow over (x)}^j→

{right arrow over (y)}^j, j=1 . . . n} is a plurality of network state transitions corresponding to the plurality of observations of the real world system;

$p<br><br>\left({\stackrel{\to <br><br>}{x}}^j\to <br><br>{\stackrel{\to <br><br>}{y}}^j<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{i=0}{\overset{N-1}{\prod <br><br>}}<br><br>\mathrm{\delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^j},y_i^j\right)$

is the likelihood function for said transition j({right arrow over (x)}^j→

{right arrow over (y)}^j;

q_i^{{right arrow over (x)}} is a binary value which determines the output of said network rule ƒ

_i to input {right arrow over (x)} wherein q_i^{{right arrow over (x)}}=1 means x_i^t=1 when {right arrow over (x)}<sup>t−

1</sup>={right arrow over (x)}, and q_i^{{right arrow over (x)}}=0 means x_i^t=0 when {right arrow over (x)}<sup>t−

1</sup>={right arrow over (x)};

Q_i={q_i^{{right arrow over (x)}}, {right arrow over (x)}ε

Z^N} wherein Z^N is the space of all binary sequences of length N;

Q={Q_i, i=0 . . . N−

1};

δ

(a, b) is the Kronecker delta function wherein δ

(a, b)=1 for a=b and 0 otherwise; and

ζ

is any additional data affecting said network model.

17. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 15 wherein each of said N network rules, ƒ

_i, determines said next state at said time step t, x_i^t, from said network state {right arrow over (x)}^τ

at said time step τ

=t−

1.

18. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 17 wherein the observations are noisy and said at least one likelihood function is defined as:

$p<br><br>\left(T<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>\underset{\stackrel{\to <br><br>}{w}\in <br><br>Z^N}{\sum <br><br>}<br><br>\left(\underset{i=0}{\overset{N-1}{\prod <br><br>}}<br><br>h<br><br>\left(q_i^{\stackrel{\to <br><br>}{w}},y_i^j\right)\right)<br><br>\left(\underset{k=0}{\overset{N-1}{\prod <br><br>}}<br><br>h<br><br>\left(w_k,x_k^j\right)\right)$

wherein

q_i^{{right arrow over (w)}} is a binary value which determines the output of said network rule ƒ

_i({right arrow over (x)}<sup>t−

1</sup>) to input {right arrow over (w)} wherein q_i^{{right arrow over (w)}}=1 means ƒ

_i^t=1 when {right arrow over (x)}<sup>t−

1</sup>={right arrow over (w)}, and q_i^{{right arrow over (w)}}=0 means ƒ

_i({right arrow over (x)}<sup>t−

1</sup>)=0 when {right arrow over (x)}<sup>t−

1</sup>={right arrow over (w)};

Q_i={q_i^{{right arrow over (w)}}, {right arrow over (w)}ε

Z^N} wherein Z^N is the space of all binary sequences of length N;

Q={Q_i, i=0 . . . N−

1};

$\underset{k=0}{\overset{N-1}{\prod <br><br>}}<br><br>h<br><br>\left(w_k,x_k^j\right)$

 

is the probability of observing input state {right arrow over (x)} given actual input state {right arrow over (w)}; and

$\underset{i=0}{\overset{N-1}{\prod <br><br>}}<br><br>h<br><br>\left(q_i^{\stackrel{\to <br><br>}{w}},y_i^j\right)$

 

is the probability of observing output state {right arrow over (y)} given actual input state {right arrow over (w)};

h(a, b) accounts for the probability of changes in said values and is defined as;

$h<br><br>\left(a,b\right)=\{\begin{array}{cc}{p}_0<br><br>\text{;<br><br>}& a=0,b=0\\ p_1<br><br>\text{;<br><br>}& a=1,b=1\\ 1-p_1<br><br>\text{;<br><br>}& a=0,b=1\\ 1-p_0<br><br>\text{;<br><br>}& a=1,b=0\end{array};<br><br>$

and

ζ

is any additional data affecting said network model.

19. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 15 wherein the observations are noisy and said at least one likelihood function is defined as:

$p<br><br>\left(T<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>p<br><br>\left(T_j<br><br>Q,\mathrm{\zeta <br><br>}\right)$

wherein;

T is a plurality of observed transition series, {T_j, j=1 . . . n};

T_j={{right arrow over (x)}<sup>t−

Tj</sup>→

{right arrow over (x)}<sup>t−

T+1j</sup>→

. . . {right arrow over (x)}^tj} is a jth one of said plurality of observed transition series;

Q_i={q_i^{{right arrow over (y)}}, {right arrow over (y)}ε

Z^NM} wherein

{right arrow over (y)} is a concatenation of said network states for said sequence of time steps having length M;

Z^NM is a space of all binary sequences of length N*M; and

q_i^{{right arrow over (y)}} is a binary value which determines the output of said network rule, ƒ

^t_i({right arrow over (x)}<sup>t−

1</sup>, {right arrow over (x)}<sup>t−

2</sup> . . . {right arrow over (x)}<sup>t−

M</sup>) to input {right arrow over (y)} wherein q_i^{{right arrow over (y)}}=1 means ƒ

^t_i=1 when ({right arrow over (x)}<sup>t−

1</sup>, {right arrow over (x)}<sup>t−

2</sup> . . . {right arrow over (x)}<sup>t−

M</sup>)={right arrow over (y)} and q_i^{{right arrow over (y)}}=0 means ƒ

_i^t= when ({right arrow over (x)}<sup>t−

1</sup>, {right arrow over (x)}<sup>t−

2</sup> . . . {right arrow over (x)}<sup>t−

M</sup>)={right arrow over (y)};

Q={Q_i, i=0 . . . N−

1} defines said N network rules;

ζ

is any additional data affecting said network model; and

P(T_j|Q, ζ

) is a probability of observing said jth transition series given said network rules.

20. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 19 wherein said probability of observing said jth transition series given said network rules is defined as:

$p<br><br>\left(T_j<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{S_j\in <br><br>Z^{\mathrm{NT}}}{\sum <br><br>}<br><br>p<br><br>\left(T_j<br><br>S_j\right)<br><br>p<br><br>\left(S_j<br><br>Q,\mathrm{\zeta <br><br>}\right)$

wherein

S is a plurality of actual transition series, {S_j, j=1 . . . n};

S_j={{right arrow over (s)}<sup>t−

Tj</sup>→

{right arrow over (s)}<sup>t−

T+1j</sup>→

. . . {right arrow over (s)}^tj};

{right arrow over (s)}^tj is a vector of actual value of said N network variables of said jth actual transition series at said time step t,

p(T_j|S_j) is the probability of said jth observed transition series, T_j given said jth actual transition series, S_j;

p(S_j|Q, ζ

) is a probability of said jth actual transition series, S_j given Q and ζ

.

21. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 20 wherein said probability of said jth observed transition series T_j given said jth actual transition series, S_j is defined as:

$p<br><br>\left(T_j<br><br>S_j\right)=\underset{\mathrm{\tau <br><br>}=t-T}{\overset{t}{\prod <br><br>}}<br><br>\underset{k=0}{\overset{N-1}{\prod <br><br>}}<br><br>h<br><br>\left(x_k^{\mathrm{\tau <br><br>}},s_k^{\mathrm{\tau <br><br>}}\right)$

wherein;

x_k^τ

is an observed value of a k^th one of said N network variables of said time step τ

;

s_k^τ

is an actual value of the k^th one of said N network variables at said time step τ

; and

h(a, b) accounts for the probability of changes in said values and is defined as;

$h<br><br>\left(a,b\right)=\{\begin{array}{cc}{p}_0<br><br>\text{;<br><br>}& a=0,b=0\\ p_1<br><br>\text{;<br><br>}& a=1,b=1\\ 1-p_1<br><br>\text{;<br><br>}& a=0,b=1\\ 1-p_0<br><br>\text{;<br><br>}& a=1,b=0\end{array}.$

22. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 20 wherein said probability of said jth actual transition series S_j given Q is defined as:

$p<br><br>\left(S_j<br><br>Q,\mathrm{\zeta <br><br>}\right)=\underset{k=0}{\overset{N-1}{\prod <br><br>}}<br><br>p<br><br>\left(s_k^{t-T,j}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>s_k^{t,j}<br><br>Q_k,\mathrm{\zeta <br><br>}\right)$

wherein

s_k<sup>t−

Tj</sup> . . . s_k^tj is said jth actual transition series at a k^th one of said N network variables; and

p(s_k<sup>t−

Tj</sup> . . . s_k^tj|Q_k, ζ

) is a probability of said jth actual transition series at said k^th network variable given Q_k and ζ

.

23. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 22 wherein said probability of said jth actual transition series at said k^th network variable given Q_k is defined as:

$p<br><br>\left(s_k^{t-T,j}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>s_k^{t,j}<br><br>Q_k,\mathrm{\zeta <br><br>}\right)=\underset{\mathrm{\tau <br><br>}=t-T+M}{\overset{t}{\prod <br><br>}}<br><br>p<br><br>\left(s_k^{\mathrm{\tau <br><br>},j}<br><br>s_k^{\mathrm{\tau <br><br>}-M,j}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>s_k^{\mathrm{\tau <br><br>}-1,j},Q_k,\mathrm{\zeta <br><br>}\right)$

wherein;

p(s_k<sup>τ

j</sup>|s_k<sup>τ

−

Mj</sup> . . . s_k<sup>τ

−

1j</sup>, Q_k, ζ

) is a probability of said actual value s_k<sup>τ

j</sup> at said k^th network variable at τ

^th one of said sequence of time steps given said actual values at said k^th network variable at M previous ones of said sequence of time steps and given Q_k.

24. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising the steps of:

defining N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

defining N network rules ƒ

_i, i=0, . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

defining N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

defining at least one likelihood function expressing the probability of making the system observations for said possible network rules; and

determining N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system;

wherein said N network variables, x_i, i=0, . . . N−

1 have m=2 values.

25. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising the steps of:

defining N network variables x_i, i=0, . . . N−

1 having values, v_i, i−

1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

defining N activation delays T_i<sup>0→

1</sup>, i=0 . . . N−

1, corresponding to said network variables x_i, i=0 . . . N−

1 from a space of possible activation delays; and

defining N deactivation delays T_i<sup>0→

1</sup>, i=0 . . . N−

1, corresponding to said N network variables x_i, i=0 . . . N−

1 from a space of possible deactivation delays;

defining N network rules ƒ

_i, i=0 . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model; and

determining at least one posterior distribution wherein said posterior distribution expresses the probabilities of said possible network rules, said possible activation delays and said possible deactivation delays given the prior knowledge and the plurality of observations of the real world system.

26. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 25 further comprising the step of defining at least one placeholder value u to represent an unknown expression level for said N network variables, x_i, i=0, . . . N−

1.

27. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 25 wherein said determining at least one posterior distribution step comprises the steps of:

defining at least one prior distribution expressing probabilities of said possible network rules, said possible activation delays and said possible deactivation delays from the prior knowledge of the real world system;

defining at least one likelihood function expressing the probability of making the system observations for each of said possible network rules, said possible activation delays and said possible deactivation delays; and

defining said at least one posterior distribution as a product of said at least one prior distribution and said at least one likelihood function.

28. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 27 further comprising the step of selecting one or more of said posterior distributions having the greatest probability to identify said network model of the real world system.

29. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 28 wherein said defining at least one prior distribution step comprises the step of defining said at least one prior probability distribution as an expression of a plurality of abstract properties of said N network rules.

30. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 29 further comprising the step of defining said space of possible network rules.

31. A method for creating a network model of a real world system of interacting components having a plurality of observations of the real world system as in claim 30 wherein said defining said space of possible network rules step comprises the steps of:

identifying at least one constraint on said space of possible network rules;

checking said space of possible network rules against said at least one constraint to identify a set of unrealistic rules; and

removing said set of unrealistic rules from consideration for faster identification of said network model of the real world system.

32. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observation of the real world system as in claim 31 wherein said at least one constraint comprises a set of limits on said plurality of abstract properties.

33. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 31 wherein said at least one constraint comprises a set of depending requirements among said N network variables, x_i, i=0, . . . N−

1.

34. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 27 further comprising the step of defining at least one placeholder value u to represent an unknown expression level for said N network variables, x_i, i=0, . . . N−

1.

35. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 27 wherein said at least one prior distribution is a uniform probability distribution.

36. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 25 further comprising the steps of:

defining a sequence of discrete time steps t, t−

1, . . . t−

T individually referenced by τ

; and

defining said network state {right arrow over (x)}^τ

of said network model at said discrete time step τ

as a vector of said values, x_i^τ

ε

{v_i, . . . v_m}, i=0, . . . N−

1 at a time step τ

.

37. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 36 further comprising the step of:

defining a plurality of network transition series T={T_j, j=1 . . . n} as T_j={{right arrow over (x)}<sup>t−

Tj</sup>→

{right arrow over (x)}<sup>t−

T+1j</sup>→

. . . {right arrow over (x)}tj} corresponding to the plurality of observations of the real world system.

38. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 37 wherein said N network rules are defined as x_i^t≈

ƒ

_i({right arrow over (x)}<sup>t−

T_i0→

1</sup>, {right arrow over (x)}<sup>t−

T_i1→

0</sup>) wherein each of said network model functions ƒ

_i determines a next state for said corresponding variable x_i at said time step t, x_i^t, from said network state {right arrow over (x)}^τ

at said time steps τ

=t−

T_i<sup>0→

1</sup>, t−

T_i<sup>1→

0</sup>.

39. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 38 wherein said at least one likelihood function is defined as:

$\begin{array}{c}p<br><br>\left(T<br><br>Q,\stackrel{\to <br><br>}{T},\mathrm{\zeta <br><br>}\right)=\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>p<br><br>\left(T_j<br><br>Q,\stackrel{\to <br><br>}{T},\mathrm{\zeta <br><br>}\right)\\ =\underset{j=1}{\overset{n}{\prod <br><br>}}<br><br>\underset{i=0}{\overset{N-1}{\prod <br><br>}}<br><br>p<br><br>\left(T_j<br><br>Q_i,T_i^{0\to <br><br>1},T_i^{1\to <br><br>0},\mathrm{\zeta <br><br>}\right)\end{array}$

wherein

p(T_j|Q_i, T_i<sup>0→

1</sup>, T_i<sup>1→

0</sup>, ζ

) is the likelihood function for said transition series T_j and said network variable x_i;

$q_i^{\stackrel{\to <br><br>}{x}}=\{\begin{array}{cc}0<br><br>\text{;<br><br>}& \left({\stackrel{\to <br><br>}{x}}^{t-T_i^{1\to <br><br>0}}=\stackrel{\to <br><br>}{x}\right)\Rightarrow <br><br>x_i^t=0\\ 1<br><br>\text{;<br><br>}& \left({\stackrel{\to <br><br>}{x}}^{t-T_i^{0\to <br><br>1}}=\stackrel{\to <br><br>}{x}\right)\Rightarrow <br><br>x_i^t=1\\ -1<br><br>\text{;<br><br>}& \mathrm{otherwise};<br><br>\end{array}$

Q_i={q_i^{{right arrow over (x)}}, {right arrow over (x)}ε

Z^N} wherein Z^N is the space of all binary sequences of length N;

Q={Q_i, i=0 . . . N−

1} ; and

ζ

is any additional data affecting said network model.

40. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 39 wherein the observations are noiseless and said likelihood function for said transition series T_j and said network variable x_i is defined as:

$\begin{array}{c}{p}_{\mathrm{noiseless}}<br><br>\left(T_j<br><br>Q_i,T_i^{0\to <br><br>1},T_i^{1\to <br><br>0)},\mathrm{\zeta <br><br>}\right)=\text{\hspace{1em}<br><br>}<br><br>\left[\underset{\mathrm{\tau <br><br>}=T_i^{0\to <br><br>1}+1}{\overset{T}{\prod <br><br>}}<br><br>\mathrm{\Delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^{\mathrm{\tau <br><br>}-T_i^{0\to <br><br>1},j}},x_i^{\mathrm{\tau <br><br>},j}\right)\right]*\\ \text{\hspace{1em}<br><br>}<br><br>\left[\underset{\mathrm{\tau <br><br>}=T_i^{1\to <br><br>0}-1}{\overset{T}{\prod <br><br>}}<br><br>\mathrm{\Delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^{\mathrm{\tau <br><br>}-T_i^{1\to <br><br>0},j}},x_i^{\mathrm{\tau <br><br>},j}\right)\right]\end{array}$

wherein $\mathrm{\Delta <br><br>}<br><br>\left(x,y\right)=\{\begin{array}{cc}1<br><br>\text{;<br><br>}& x=-1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{or}<br><br>\text{\hspace{1em}<br><br>}<br><br>y=-1\\ \mathrm{\delta <br><br>}<br><br>\left(x,y\right)<br><br>\text{;<br><br>}& \mathrm{otherwise}.\end{array}$

41. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 39 wherein the observations are noisy further comprising the step of:

defining a plurality of actual network transition series S={S_i, j=1 . . . n} .

42. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 41 wherein said likelihood function for said transition series T_j and said network variable x_i is defined as:

$p_{\mathrm{noise}}<br><br>\left(T_j<br><br>Q_i,T_i^{0\to <br><br>1},T_i^{1\to <br><br>0},\mathrm{\zeta <br><br>}\right)=\underset{S_j}{\sum <br><br>}<br><br>p<br><br>\left(T_j<br><br>S_j\right)<br><br>p_{\mathrm{noiseless}}<br><br>\left(S_j<br><br>Q_i,T_i^{0\to <br><br>1},T_i^{1\to <br><br>0},\mathrm{\zeta <br><br>}\right)$

wherein $\begin{array}{c}{p}_{\mathrm{noiseless}}<br><br>\left(S_j<br><br>Q_i,T_i^{0\to <br><br>1},T_i^{1\to <br><br>0},\mathrm{\zeta <br><br>}\right)=\text{\hspace{1em}<br><br>}<br><br>\left[\underset{\mathrm{\tau <br><br>}=T_i^{0\to <br><br>1}-1}{\overset{T}{\prod <br><br>}}<br><br>\mathrm{\Delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^{\mathrm{\tau <br><br>}-T_i^{0\to <br><br>1},j}},w_i^{\mathrm{\tau <br><br>},j}\right)\right]*\\ \text{\hspace{1em}<br><br>}<br><br>\left[\underset{\mathrm{\tau <br><br>}=T_i^{1\to <br><br>0}+1}{\overset{T}{\prod <br><br>}}<br><br>\mathrm{\Delta <br><br>}<br><br>\left(q_i^{{\stackrel{\to <br><br>}{x}}^{\mathrm{\tau <br><br>}-T_i^{1\to <br><br>0},j}},w_i^{\mathrm{\tau <br><br>},j}\right)\right];<br><br>\end{array}$$\mathrm{\Delta <br><br>}<br><br>\left(x,y\right)=\{\begin{array}{cc}1<br><br>\text{;<br><br>}& x=-1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{or}<br><br>\text{\hspace{1em}<br><br>}<br><br>y=-1\\ \mathrm{\delta <br><br>}<br><br>\left(x,y\right)<br><br>\text{;<br><br>}& \mathrm{otherwise};<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}\end{array}$

p(T_j|S_j) relates said actual network transition series to said network transition series corresponding to the plurality of observations of the real world system; and

w_i<sup>τ

j</sup> is the actual value of said network variable x_i at said time step τ

of said actual transition series S_j.

43. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 25 wherein said N network variables, x_i, i=0, . . . N−

1 have m=2 values.

44. A method for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system as in claim 25 wherein the network model comprises at least one of a multi-level logic network model, a boolean network model or a genetic regulatory network.

45. Computer executable software code stored on a computer readable medium, the code for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system, the code comprising:

code to define N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

code to define N network rules ƒ

_i, i=0, . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

code to define N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

code to define at least one likelihood function expressing the probability of making the system observations for said possible network rules;

code to determine N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system; and

code to select one or more of said posterior distributions having the greatest probability to identify said network model of the real world system;

wherein said code to define N prior distributions step comprises code to define said N prior probability distributions as an expression of a plurality of abstract properties of said N network rules.

46. A programmed computer for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising a memory having at least one region storing computer executable program code and a processor for executing the program code stored in said memory, wherein the program code includes:

code to define N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

code to define N network rules ƒ

_i, i=0, . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model;

code to define N prior distributions corresponding to said N network rules expressing probabilities of said possible network rules from the prior knowledge of the real world system;

code to define at least one likelihood function expressing the probability of making the system observations for said possible network rules;

code to determine N posterior distributions defined as a product of said prior distributions and said at least one likelihood function wherein said posterior distributions express the probabilities of said possible network rules given the prior knowledge and the plurality of observations of the real world system; and

code to select one or more of said posterior distributions having the greatest probability to identify said network model of the real world system;

wherein said code to define N prior distributions step comprises code to define said N prior probability distributions as an expression of a plurality of abstract properties of said N network rules.

47. Computer executable software code stored on a computer readable medium, the code for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system, the code comprising:

code to define N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

code to define N activation delays T_i<sup>0→

1</sup>, i=0 . . . N−

1, corresponding to said N network variables x_i, i=0 . . . N−

1 from a space of possible activation delays; and

code to define N deactivation delays T_i<sup>1→

0</sup>, i=0 . . . N−

1, corresponding to said N network variables x_i, i=0, . . . N−

1 from a space of possible deactivation delays;

code to define N network rules ƒ

_i, i=0, . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model; and

code to determine at least one posterior distribution wherein said posterior distribution expresses the probabilities of said possible network rules, said possible activation delays and said possible deactivation delays given the prior knowledge and the plurality of observations of the real world system.

48. A programmed computer for creating a network model of a real world system of interacting components having a plurality of expression levels from prior knowledge and a plurality of observations of the real world system comprising a memory having at least one region storing computer executable program code and a processor for executing the program code stored in said memory, wherein the program code includes:

code to define N network variables x_i, i=0, . . . N−

1 having values, v_i, i=1, . . . m to represent the components of the real world system, said values defining a network state of said network model;

code to define N activation delays T_i<sup>0→

1</sup>, i=0 . . . N−

1, corresponding to said N network variables x_i, i=0 . . . N−

1 from a space of possible activation delays; and

code to define N deactivation delays T_i<sup>0→

1</sup>, i=0 . . . N−

1, corresponding to said N network variables x_i, i=0 . . . N−

1 from a space of possible deactivation delays;

code to define N network rules ƒ

_i, i=0 . . . N−

1 corresponding to said N network variables from a space of possible network rules wherein said N network rules have outputs defining said network model; and

code to determine at least one posterior distribution wherein said posterior distribution expresses the probabilities of said possible network rules, said possible activation delays and said possible deactivation delays given the prior knowledge and the plurality of observations of the real world system.