Reduction of Boolean network models Elena Dimitrova School of Mathematical and Statistical Sciences Clemson University http://edimit.people.clemson.edu/ Algebraic Biology E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 1 / 18
Motivation In the previous lecture, we modeled time-delays and dilution & degradation by adding a number of Booleans variables. This can causes the state space to grow enormously, though in many cases, this shouldn’t affect the qualitative nature of the dynamics. In other cases, certain Boolean network models are huge and too big for direct analysis. In this lecture, we’ll see how large Boolean networks can be “reduced” to much smaller models in a way that preserves the key feature such as fixed points. E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 2 / 18
Wiring diagrams Definition A Boolean network (BN) in the Boolean variables x 1 , . . . , x n is a function f “ p f 1 , . . . , f n q : F n Ñ F n 2 Ý 2 where each f i : F n 2 Ñ F 2 is called a coordinate or local function. Definition The wiring diagram of a Boolean network is a directed graph G on vertex set x 1 , . . . , x n (or just 1 , . . . , n ) and a directed edge p x i , x j q if f j depends on x i . An edge x i Ý Ñ x j is positive if f j p x 1 , . . . , x i ´ 1 , 0 , x i ` 1 , . . . , x n q ď f j p x 1 , . . . , x i ´ 1 , 1 , x i ` 1 , . . . , x n q and negative if the inequality is reversed. Negative edges are denoted with circles or blunt arrows instead of traditional arrowheads. Definition A Boolean function f i is unate (or monotone) if every edge in the wiring diagram is either positive or negative. E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 3 / 18
Wiring diagrams x j A positive edge x i represents a situation where i activates j . Examples. f j “ x i ^ y : 0 “ f j p x i “ 0 , y q ď f j p x i “ 1 , y q ď 1. f j “ x i _ y : 0 ď f j p x i “ 0 , y q ď f j p x i “ 1 , y q “ 1. A negative edge x i x j represents a situation where i inhibits j . Examples. f j “ x i ^ y : 1 ě f j p x i “ 0 , y q ě f j p x i “ 1 , y q “ 0. f j “ x i _ y : 1 “ f j p x i “ 0 , y q ě f j p x i “ 1 , y q ě 0. Occasionally, edges are neither positive nor negative: Example. (The logical “XOR” function): 0 “ f j p x 1 “ 0 , y “ 0 q ă f j p x 1 “ 1 , y “ 0 q “ 1 f j “ p x i ^ y q _ p x i ^ y q : 1 “ f j p x 1 “ 0 , y “ 1 q ą f j p x 1 “ 1 , y “ 1 q “ 0 Most edges in real Boolean models are either positive or negative because most biological interactions are either simple activations or inhibitions. E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 4 / 18
A motivating example Toy model of the lac operon f M “ R R represses mRNA production f P “ M P is produced by translation of mRNA f B “ M B is produced by translation of mRNA f R “ A A inactivates the repressor protein f A “ L ^ B A is produced by lactose and β -galactosidase f L “ P Lac permease transports lactose into the cell Here is the wiring diagram: M B R P L A We won’t show the state space because it’s large (64 nodes), but it has two fixed points, both of which are biologically reasonable: p M , P , B , R , A , L q “ p 0 , 0 , 0 , 1 , 0 , 0 q and p 1 , 1 , 1 , 0 , 1 , 1 q . Our goal is to “reduce” this model in a way that in some senes, preserves the fixed points . E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 5 / 18
A motivating example (cont.) Toy model of the lac operon f M “ R f P “ M M B R f B “ M f R “ A f A “ L ^ B P L A f L “ P Consider the variable P . At equilibrium, P p t q “ P p t ` 1 q “ f P p x p t qq “ M p t q . Similarly, we can conclude that B p t q “ B p t ` 1 q “ f B p x p t qq “ M p t q . Thus, we can replace every instance of P and B with M : f M “ R f P “ M M R f B “ M f R “ A f A “ L ^ M f L “ M L A There are two steady-states of this reduced network: p M , R , A , L q “ p 0 , 1 , 0 , 0 q , p 1 , 0 , 1 , 1 q . Moreover, since B “ M , P “ M , we can recover the steady-states of the original network. E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 6 / 18
A motivating example (cont.) Partially reduced model of the lac operon f M “ R M R f R “ A f A “ L ^ M f L “ M L A We can reduce further. At equilibrium, A “ f A “ L ^ M , so we can replace every instance of A with L ^ M : f M “ R M R f R “ L ^ M “ L _ M f A “ L ^ M f L “ M L There are two fixed points of this reduced network: p M , R , L q “ p 0 , 1 , 0 q , p 1 , 0 , 1 q . Moreover, since B “ P “ M , A “ L ^ M , we can recover the fixed points of the original network by back-substituting . p M , P , B , R , A , L q “ p M , M , M , R , L ^ M , L q “ p 0 , 0 , 0 , 1 , 0 , 0 q , and p 1 , 1 , 1 , 0 , 1 , 1 q . E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 7 / 18
A motivating example (cont.) Partially reduced model of the lac operon f M “ R M R f R “ L _ M f L “ M L We can reduce further. At equilibrium, L “ f L “ M , so we can replace every instance of L with M : f M “ R f R “ M _ M “ M M R f L “ M There are two fixed points of this reduced network p M , R q “ p 0 , 1 q , p 1 , 0 q . Moreover, since L “ B “ P “ M and A “ L ^ M “ M , we can recover the steady-states of the original network by back-substituting . p M , P , B , R , A , L q “ p M , M , M , R , M , M q “ p 0 , 0 , 0 , 1 , 0 , 0 q , and p 1 , 1 , 1 , 0 , 1 , 1 q . E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 8 / 18
General reduction Reduction steps 1. Simplify the Boolean functions and wiring diagram. 1.1 Reduce / simplfy Boolean expressions using Boolean algebra. 1.2 Remove unnecessary edges from the wiring diagram. 2. Delete vertices x i with no self-loop (equivalently, f x i doesn’t depend on x i ), by doing the following: 2.1 For all vertices y such that x i Ý Ñ y , substitute f x i into x i : f y p x 1 . . . , ¨ ¨ ¨ x i ¨ ¨ ¨ , . . . , x n q becomes f y pp x 1 . . . , ¨ ¨ ¨ f x i ¨ ¨ ¨ , . . . , x n q . looomooon loooomoooon pos. x i pos. x i 2.2 Replace edges v Ý Ñ x i Ý Ñ y by v Ý Ñ y and remove x i (and all edges to/from x i ). Exercise (HW) In Step 2.2 above, how should you replace replace edges of the form: y v x i x i y v x i y v E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 9 / 18
General reduction: an example 1 2 Consider the Boolean network f p x q “ p x 2 , p x 1 ^ x 3 q _ x 2 , x 1 q . 3 Let’s remove x 3 “ x 1 . The new Boolean functions are h 1 p x 1 , x 2 q “ f 1 p x 1 , x 2 , x 3 q “ f 1 p x 1 , x 2 , x 1 q “ x 2 , h 2 p x 1 , x 2 q “ f 2 p x 1 , x 2 , x 3 q “ f 2 p x 1 , x 2 , x 1 q “ p x 1 ^ x 1 q _ x 2 However, x 1 ^ x 1 “ 0, and so h 2 p x 1 , x 2 q “ p x 1 ^ x 1 q _ x 2 “ 0 _ x 2 “ x 2 . 1 2 The reduced Boolean network is thus h p x 1 , x 2 q “ p x 2 , x 2 q To find the fixed points, we must solve the system h i “ x i for i “ 1 , 2: " h 1 p x 1 , x 2 q “ x 2 “ x 1 h 2 p x 1 , x 2 q “ x 2 “ x 2 . Since x 2 ‰ x 2 , there are no fixed points in the reduced BN, and thus none in the original BN. E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 10 / 18
General reduction: an example 1 2 3 Consider the Boolean network: f “ p x 5 _ x 2 _ x 4 , x 1 ^ x 3 , x 2 , x 2 , x 1 _ x 4 q . 5 4 Remove x 5 “ x 1 _ x 4 : 1 2 3 f “ pp x 1 _ x 4 q _ x 2 _ x 4 , x 1 ^ x 3 , x 2 , x 2 q “ p x 1 _ x 2 _ x 4 , x 1 ^ x 3 , x 2 , x 2 q . 4 Remove x 4 “ x 2 : 1 2 3 f “ p x 1 _ x 2 _ x 2 , x 1 ^ x 3 , x 2 q “ p x 1 _ x 2 , x 1 ^ x 3 , x 2 q Remove x 3 “ x 2 : 1 2 f “ p x 1 _ x 2 , x 1 ^ x 2 q “ p x 1 _ x 2 , x 1 ^ x 2 q This yields the system: The reduced system p h 1 , h 2 q has 2 fixed points: h 1 p x 1 , x 2 q “ x 1 _ x 2 $ p x 1 , x 2 q “ p 1 , 0 q , p 0 , 1 q . ’ h 2 p x 1 , x 2 q “ x 1 ^ x 2 ’ ’ & x 3 “ x 2 Thus, the original system has two fixed points: x 4 “ x 2 ’ ’ ’ % p x 1 , x 2 , x 3 , x 4 , x 5 q “ p 1 , 0 , 1 , 1 , 1 q , p 0 , 1 , 0 , 0 , 0 q . x 5 “ x 1 _ x 4 E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 11 / 18
Computational algebra software: Macaulay2 and Sage Macaulay2 is a free computer algebra system developed by Dan Grayson (UIUC) and Mike Stillman (Cornell). It is named after the English mathematician Francis Macaulay (1862–1937). It can be downloaded or used online at www.math.uiuc.edu/Macaulay2 . Alternatively, it has been incorporated into the Sage Math Cloud: https://cloud.sagemath.org . Let’s see how to use Macaulay2 in Sage to do the Boolean reduction from the previous slide. First, tell Sage that we want to use Macaulay2 (hit Shift-Enter after each command): %default_mode macaulay2 We want polynomials in variables x 1 , . . . , x 5 , over the field F 2 , and x 2 i “ x i : R = ZZ/2[x1,x2,x3,x4,x5] / ideal(x1^2-x1, x2^2-x2, x3^2-x3, x4^2-x4, x5^2-x5); For convenience, let’s define a | b : “ a ` b ` ab and a & b : “ a ˚ b : RingElement | RingElement :=(x,y)->x+y+x*y; RingElement & RingElement :=(x,y)->x*y; E. Dimitrova (Clemson) Reduction of Boolean network models Algebraic Biology 12 / 18
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