4.2 PNFL Models Can Mimic Common Network Motifs
4.2.1 Feed-Forward Loop
A feed-forward loop is composed of three genes: A transcription factorFregulates a co-factorC, which both jointly regulate a target geneT. Thus, the feed-forward loop has three regulatory in- teractions (FtoC,FtoT,CtoT) which can be either activating or inhibiting, allowing for eight different combinations. The effects ofFandCare integrated at the cis-regulatory elements ofT. Two simple types of integration are AND-like or OR-like functions. Thus, 16 different configu- rations of regulatory effects and integrations are possible. Mangan and Alon [110] performed a theoretical analysis of these 16 types of feed-forward loops. They found that feed-forward loops can accelerate or delay a signal transduction, and serve as persistence detectors or pulsers, de- pending on the configuration of the regulatory interactions and integration. The most abundant type of feed-forward loop found inE. coli as well as inS. cerevisiaehas three activating regu- latory interactions and an AND-like integration [110]. This feed-forward loop causes a delay in initial signal transduction. Target geneT is not expressed immediately after its direct regulator
F is inserted into the system, but only after co-factorChas reached a minimum expression level. On the other hand, expression level of T drops without delay as soon asF is removed. Due to the initial delay, the feed-forward loop can act as a persistence detector. A short pulse ofFis not
4.2 PNFL Models Can Mimic Common Network Motifs 53 Mk+1(P) =Mk(P) +fA(A), fA(A) =−Mk(P) + f lsA(A) ⇒ Mk+1(P) = f lsA(A) 𝒇𝒍𝒔𝑨 A absent present 0 1 1 k+1 0 P k iterations
figure_PNFL_semiqual_3
fB(A) =−Mk(P) + f lsPA(P,A) ⇒ Mk+1(P) = f lsPA(P,A) 𝒇𝒍𝒔𝑷𝑨 Plow medium high
A ab se n t 0 0 0.5 p re se n t 0.5 1.0 1.0 1 k+1 0 P k iterations slowed change k+2 0.5
figure_PNFL_semiqual_5
fC(A) =−ω·Mk(P) +ω·f lsA(A) ⇒ Mk+1(P) = (1−ω)·Mk(P) +ω·f lsA(A) 1 k+1 0 P k iterations slowed change k+2 … 0.5figure_PNFL_semiqual_6
Figure 4.5 Modeling of slow state changes.If the semi-discrete modeling approach is applied, fuzzy
logic systems are designed to propose target markings for places. Repeated firing of transitions must result in a convergence to the proposed target markings. If the functions that are assigned to arcs are designed such that they fully replace the current marking, target markings are reached after a single iteration (top). Here, fuzzy logic system f lsA proposes a new marking for place P depending on the state ofA. If the state ofAswitches from absent to present at iterationk, then function fA causes that the marking of P switches from 0 to 1 in one iteration. To slow down the marking changes, one could enforce intermediate steps by taking the current marking of the target entity in consideration (center). Fuzzy logic system f lsPA depends on the current states ofPandA. IfAispresent, the state ofPis increased by one step, fromlow
tomedium, ormediumtohigh. Hereby, the centers of gravity oflow,medium, andhighare 0, 0.5, and 1. If the state ofAswitches fromabsent to presentat iterationk, then function fB causes the marking ofP to switch from 0 to 1 in two iterations. As any further slowdown by one iteration can only be achieved by adding an antecedent fuzzy set and adding several rules, this approach is extremely inefficient. A more efficient approach is to design functions such that they do not fully replace the old marking, but instead calculate a weighted mean between old marking and the marking proposed by the fuzzy logic system (bottom). Function fCcalculates a weighted average based on an update-factorω∈(0,1].
54 4. Joining Petri Nets and Fuzzy Logic: PNFL Modeling
sufficient to cause a strong expression ofT, while it is sufficient to cause a considerable response in expression ofC.
We created a PNFL model of this type of feed-forward loop using the semi-discrete modeling approach (Figure 4.6). Thus, fuzzy logic systems assigned to arcs compute new states for target places. The current states of targets are used as inputs to fuzzy logic systems. Rules were defined such that only one-step changes of the target entities state are allowed, e.g. a state change from
lowtomediumis allowed, but not fromlowtohigh. This slows down the state changes. Fuzzy
logic system f lsC(F,C) represents a simple activating function, that increases expression ofC
only ifF is presentand decreases it otherwise:
R1: IF F is present AND C is low T HEN y1is0.5
R2: IF F is present AND C is medium T HEN y2is1.0
R3: IF F is present AND C is high T HEN y3is1.0
R4: IF F is absent AND C is low T HEN y4is0.0
R5: IF F is absent AND C is medium T HEN y5is0.0
R6: IF F is absent AND C is high T HEN y6is0.5
The consequent fuzzy sets are identical to the antecedent fuzzy setlow, medium, andhigh and are represented by their centers of gravity in the aforementioned rules.
Three experimental setting were simulated:
1. Prolonged pulse.Expression level ofF is set to 1 and subject to exponential decay with rate 0.005. Expression levels ofC andT increase to the maximum level and persist. The initial expression ofT is delayed as compared toC.
2. Long pulse.Expression level of F is set to 1 and subject to exponential decay with rate 0.05. Expression levels ofCandT increase to the maximum level. WhenF drops below 70 % of its maximum, bothCandT expression levels start to decrease and drop to zero. 3. Short pulse.Expression level ofF is set to 1 and subject to exponential decay with rate
0.5. Expression ofCincreases to 50 % of its maximum and drops to zero immediately after expression ofF drops below 50 %. Expression ofT increases only slightly to about 12 % before dropping to zero again.
The simulations show that the PNFL model of the feed-forward loop with three activating regula- tory interactions and AND-like integration behaves qualitatively similar to the theoretical results of Mangan and Alon [110]. I.e. we observe a delay of the target gene’s initial expression and observe that short signal pulses are filtered and do not cause a strong reaction of the target gene. Fuzzy set definitions can be modified to change the quantitative behavior of the system. For example, fuzzy sets absent and present can be modified such that given the same signal, the response of the system is prolonged or shortened, or such thatCandPdecrease faster or slower. We will discuss this in more detail in the following. Fuzzy setsabsent and present are realized as unbounded trapezoidal fuzzy sets and are defined by points L=0.3 (right top of absent and left border of present) and R=0.7 (right border of absent and left top of present). A signal
4.2 PNFL Models Can Mimic Common Network Motifs 55
concentration smaller than L is considered as insufficient to trigger a response, a signal larger than R triggers the maximal response, and a signal between L and R triggers an intermediate response. We consider two special iterations in the time course of P. I1 is the first iteration
wherePdrops below the maximal value that was attained after the signal triggered, andI2is the first iteration wherePdrops to zero again. We consider four modified definitions ofabsent and
present:
1. The distance betweenLandRis shortened. This causesI1to be shifted to the right andI2
to be shifted to the left, The result is a prolonged maximal response to the signal, but the response drops more rapidly when the signal fades.
2. The distance betweenLandRis widened. This causesI1to be shifted to the left andI2 to be shifted to the right. The systems response starts to drop early but reaches zero later. 3. LandRare shifted to the left.I1andI2 are shifted to the right. The system already reacts
to small signal concentrations.
4. LandRare shifted to the right.I1 andI2 are shifted to the left. The system reacts to high signal concentrations only.
None of the discussed modifications to fuzzy setsabsent and presentinfluences the overall qua- litative behavior, i.e. the delay of signal transduction and the filtering of signal bursts, although the maximal length of filtered bursts may vary.