Biological system modelling is used to guide experimental work, therefore reducing the time and cost of in vivo implementation of newly designed systems. We introduce an improved modelling method, based on fuzzy logic and Petri nets. By using fuzzy logic to linguistically describe a biological process, we avoid the necessity to use kinetic rates, which are often unknown. We introduce a new set of transition functions to enable the use of our method with existing Continuous Petri nets. With this we achieve the extension of usability and applicability of current Continuous Petri nets de nition even for biological systems for which exact kinetic data are unknown. We demonstrate the contribution of our approach by using it to model the translation in a simple transcription-translation system. We compare the results obtained to the results of exiting ODE approaches.
Advances in synthetic biology are consistently opening new possibilities for the
design and construction of complex biological systems. Because in vivo design
is costly and time-consuming, various modelling methods can be used to check
whether the desired behaviour of the system is achievable in silico rst [1{3].
Furthermore, modelling enables us to test in what way small or substantial changes
to the design of our system a ects its behaviour and potentially change the design
before implementing it in vivo. Which modelling method to use depends on the
size of the system, the desired accuracy of simulation results and whether
accurate kinetic rates, which determine system's dynamics, are known [
rst step of the design process presented on the
diagram is only possible when we are building a model with well characterized parts
(left side), while our approach can be used for modelling biological systems in
the same way even if accurate kinetic data is unknown (right side). Existing
methods are often used within the framework of Petri nets, a formalism that
has been extended to suit the needs for continuous deterministic and stochastic
approaches [
Fuzzy logic quantitative modelling approach n g i s e d t n e r e p x e d e d i u m i G 2
Experimental results * * * * ** ** * * * * ** * ** * * * ** * * ODE and Stochastic model ing approaches iif c a it o n a n d if n e t u n i n g
Desired behavior 1 1 3 g iifrc
Experimental results * * * * ** ** * * * ** * * * ** * ** * * * Fuzzy model ing approach 2 G u i d e d e x p e r i m e n t d e s i g n
Model design ( ) approaches are often not usable (left side). Proposed modelling method uses the same paradigm for model building (right side), but can be used even when accurate kinetic rates are unknown.
Similarly to quantitative Petri nets, fuzzy logic Petri nets have been established
as a very promising modelling approach for qualitative analysis of biological
systems. Fuzzy logic uses linguistic terms and rules for system behaviour
description, allowing intuitive design and model construction. It has been applied
to several research areas such as: extracting activator/repressor relationship from
micro-array data [
This paper is organized as follows: in Section 2 we present the basics of fuzzy logic modelling and how fuzzy logic is used in the Petri net framework. In Section 3 we demonstrate the proposed method by constructing a model of translation in a simple transcription-translation system, in Section 4 simulation results obtained with ODE and proposed method are compared and in Section 5 we summarize what the main contribution of the method is and give some directions for future research. 2 2.1
Fuzzy logic uses linguistic terms and rules to describe current system state and
how the state of the system changes over time [
By using Petri net formalism it is possible to intuitively build the Petri net
graph of the model. Once the Petri net is constructed using di erent modelling
methods is easy. We only need to change the underlying transition function and
ring rules. Continuous Petri nets use real numbers in places (marking values),
meaning that transitions also no longer consume and produce whole tokens, but
instead change the marking of an input or output place by a real value. New
marking values in places are calculated by adding the contribution from input
transitions and subtracting the value that is consumed due to output transitions.
This allows a continuous ow throughout the Petri net, which can be used to
present a system of ODEs [
A(x) = 8 x a >< cb xa
c b >:0 a x b; b x c; otherwise; y =
Pn i=1 yi Pn i=1 [i] [i] ; (1) (2) where y is the crisp value, yi x-coordinate at which membership function of fuzzy set i has the highest possible degree of membership (parameter b from Eqn. 1) and [i] current degree of membership for fuzzy set i. Output fuzzy value is obtained by applying IF-THEN rules to the input variables. With basic (one input and one output) IF-THEN rules fIF T HEN is simple. If we have an input variable x, an output variable y and a rule IF x is Low THEN y is High, x membership degree to its fuzzy set Low is assigned to y membership degree to its fuzzy set High. This process is then repeated for all rules to obtain fuzzy value of y. However, biological processes usually have more than one input chemical species, therefore we need to use rules with more than one input variable. When applying IF-THEN rules with more than one input variables we usually de ne the rules using operator AND, which acts as a function M in( 1[i]; 2[i]; :::; n[i]), where j [i] is a membership degree of variable j to its fuzzy set i. If we have two input variables x1, x2, an output variable y and a rule IF x1 is Low AND x2 is High THEN y is High, y degree of membership to its fuzzy set High would be assigned as a lower of the two values: x1 degree of membership to its fuzzy set Low and x2 membership degree to its fuzzy set High, which we can also note as y[High] = M in( x1 [Low]; x2 [High]). Once we de ne the set of these three types of functions (fuzzi cation, defuzzi cation, IF-THEN rules), we have all the tools needed to construct a fuzzy Petri net model of a biological process.
We present model construction using proposed method on a simple
transcriptiontranslation system introduced in [
We will adopt the ODE model of this system from [
mRNA [mRN A]; d[T sR] dt = kT sR [T sR] [DN A] mts + [DN A]
; d[GF P ] dt = ktl [T lR] [mRN A] mtl + [mRN A]
kmat [GF P ]; d[T lR] dt =
T lR [T lR] : mT lR + [T lR] (3) (4) (5) (6)
The ODE model from [
According to [
Using this constructed Petri net, we will observe how concentration of GFP changes over time and when it reaches its maximum value if we add the DNA at di erent time points and compare the simulation results to the ODE model. 4
Both ODE and Fuzzy logic models were built in MATLAB Simulink. Petri nets serve as a powerful framework for both approaches, however computing underlying numerical solutions can be done by an external engine like MATLAB. We Fuzzification
VeryHigh 0.72
IF-THEN rules
Defuzzification mRNA concentration
970 TlR concentration 0.83
. . .
used MATLAB Simulink built-in ode4 (Runge-Kutta) solver and set the
simulation time to 1000 minutes with a 0.1 minute xed time step. Initial concentrations
of both TsR and TlR were set to 1 nM, all others were set to 0 nM. During the
simulation we inserted 3.4 nM of DNA at 6 di erent time points (six di erent
simulations with same initial concentrations): 0 minutes, 37 minutes, 73
minutes, 112 minutes, 153 minutes and 187 minutes (these concentrations and time
points were chosen according to [
Simulation results from both models show that the plateau of protein concentration is reached at the same time (at about 200 minutes) which is the result of translation resource degrading to 0, stopping translation entirely. Since we did not include protein degradation, its concentration stays unchanged for the remaining time of simulation. We see that even though we described translation with fuzzy approach we still get comparable quantitative results. The error introduced due to using rough estimation of translation speed instead of exact translation rate is noticeable. However, we did not use any exact parameter values for translation with the proposed method and still managed to obtain quantitatively and biologically relevant results, which are comparable to those obtained with (strict) ODE approach. In addition, because we only changed how we model translation, trajectories for other processes stay unchanged. Simulation results indicate that fuzzy logic is a viable modelling approach even when kinetic data is unknown. By exploiting information we have about the system for similar models and biological systems, we can successfully build a quantitative model even when accurate parameters are unknown. By using our approach with Petri nets, we can easily change the underlying description of a process for which kinetic data is unknown while preserving accuracy of ODEs for the parts of system where it is possible. 5
We presented the Fuzzy logic approach for modelling biological processes, which avoids using exact kinetic data. Proposed method uses a rough estimation of process dynamic to obtain quantitative simulation results. This estimation is extracted from existing base of knowledge about modelling biological processes by inspecting similar systems and chemical species. With introducing this method to Petri nets we managed to further extend their usability and applicability to continuous approaches, even when kinetic data is unknown. We showed its uses on a simple transcription-translation system by substituting the ODE translation description with the proposed fuzzy approach, achieving quantitatively and biologically relevant results, without using exact kinetic data. Adding additional functions for fuzzi cation, application of IF-THEN rules and defuzzi cation increases the complexity of the Petri net model. However, these functions are very simple and can be evaluated the same way that ODEs are. In addition, these three stages of fuzzy logic are repeated for every process for which we use the proposed approach and while we need to manually de ne fuzzy sets, membership functions and IF-THEN rules, once those are de ned we could generate the Fuzzy Petri net automatically. The number of transitions and edges for fuzzi cation and defuzzi cation stages are de ned by the number of fuzzy sets, while the functions for these transitions are de ned by the shape of membership functions. Number of edge and transition functions in IF-THEN rule stage are de ned by IF-THEN rules (e.g. IF x1 is High AND x2 is Low THEN y is Low would generate a transition with two input edges - from places x1High and x2Low - and one output edge - to place yLow; the function in the transition would be Min(Input 1, Input 2)). Moreover, we could use hierarchical Petri net structure, where top level would resemble the Petri net shown on Figure 3, while the fuzzi cation, IF-THEN rules and defuzzi cation stages (Figure 6) would be presented as a lower level Fuzzy Petri net that describes all three stages as one transition (in our case translation). Our future research also includes using this approach on a more complex system and observe how inaccuracy of our rough estimation changes the overall trajectory of concentrations. We would also like to consider using experimental data for ne tuning our estimations, which would bring the accuracy of simulation results even closer to those of existing methods.
Results presented here are in scope of PhD thesis that is being prepared by Jure Bordon, University of Ljubljana, Faculty of Computer and Information science.