# Shaping and Dilating the Fitness Landscape for Parameter Estimation in Stochastic Biochemical Models

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## Abstract

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## Featured Application

**Potential applications of this work regard the parameter estimation and the resulting analyses of complex biochemical systems characterized by a stochastic behavior, which allows for elucidating the unknown underlying mechanisms.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reaction-Based Modeling and Stochastic Simulation Algorithm

- The set $\mathcal{S}=\{{S}_{1},\dots ,{S}_{N}\}$ of molecular species;
- The set $\mathcal{R}=\{{R}_{1},\dots ,{R}_{M}\}$ of the biochemical reactions describing all interactions among the species in $\mathcal{S}$.

- ${R}_{1}:S+E\stackrel{{c}_{1}={10}^{-2}}{\to}ES$;
- ${R}_{2}:ES\stackrel{{c}_{2}={10}^{-1}}{\to}S+E$;
- ${R}_{3}:ES\stackrel{{c}_{3}=1}{\to}E+P$.

#### 2.2. Fuzzy Self-Tuning Particle Swarm Optimization

#### 2.3. Dilation Functions

#### 2.4. Evolving Dilation Functions

#### 2.5. Surrogate Fourier Modeling with surF

- 1.
**Discrete Cosine Transform.**

- 2.
**Reducing the number of samples.**

- If $\overrightarrow{z}$ is in the convex hull defined by the points ${\overrightarrow{z}}_{0},\dots {\overrightarrow{z}}_{\sigma -1}$, then $\widehat{f}\left(\overrightarrow{z}\right)$ is obtained by a linear interpolation;
- otherwise, a linear interpolation is not possible and $\widehat{f}\left(\overrightarrow{z}\right)$ is defined as $\widehat{f}\left({\overrightarrow{z}}^{\prime}\right)$, where ${\overrightarrow{z}}^{\prime}$ is the point among ${\overrightarrow{z}}_{0},\dots {\overrightarrow{z}}_{\sigma -1}$ nearest to $\overrightarrow{z}$.

- 3.
**Parameters of surF.**

- $\sigma $, which is the number of samples from f used to build $\widehat{f}$;
- $\rho $, which is the “density” of samples from $\widehat{f}$ to obtain the ${\rho}^{D}$ points used to calculate the DFT; and
- $\gamma $, which controls the number of low frequencies preserved.

## 3. Results

#### 3.1. Effect of DFs on the PE Problem

#### 3.2. Combining DFs and Fourier Surrogate Modeling

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ABF | Average Best Fitness |

BF | Basis Function |

DCT | Discrete Cosine Transform |

DF | Dilation Function |

E | Enzyme |

ES | Enzyme–Substrate complex |

FRBS | Fuzzy Rule-Based System |

FST-PSO | Fuzzy Self-Tuning Particle Swarm Optimization |

MM | Michaelis–Menten |

P | Product |

PE | Parameter Estimation |

PSO | Particle Swarm Optimization |

RBM | Reaction-Based Model |

S | Substrate |

SSA | Stochastic Simulation Algorithm |

surF | Fitness Landscape Surrogate Modeling with Fourier Filtering |

Mathematical Notation | |

${\alpha}_{m,n}$ | stoichiometric coefficients associated with the n-th reactant |

${\beta}_{m,n}$ | stoichiometric coefficients associated with the m-th reaction |

$\mathbf{c}$ | vector of stochastic constants |

${c}_{m}$ | stochastic (kinetic) constant |

${C}_{\mathrm{cog}}$ | cognitive attractor of FST-PSO |

${C}_{\mathrm{soc}}$ | social attractor of FST-PSO |

D | number of dimensions of the search space |

${d}_{m}$ | number of distinct combinations of the reactant molecules |

f | original fitness function |

$\tilde{f}$ | dilated fitness function |

$\overline{f}$, $\widehat{f}$ | surrogate fitness functions |

${\mathcal{F}}_{r}\left({q}_{p}\right)$ | folding operator |

$\gamma $ | number of lower frequencies to not be zeroed |

I | number of sampled points to compute the dilated landscape |

${[\ell ,u]}^{D}$ | lower and upper bounds of the search space |

${l}_{p}$ | linear basis function |

${O}_{n}\left({t}_{k}\right)$ | the experimental (target) amount of ${S}_{n}$ measured at time ${t}_{k}$ |

p | parameter of the linear basis function ${l}_{p}$ |

${\psi}_{\rho}$ | coefficient representing the amplitude of $\rho $-th frequency |

r | parameter of the folding operator |

$\mathcal{R}$ | set of biochemical reactions |

${R}_{m}$ | m-th biochemical reaction |

$\rho $ | number of equally spaced points to build the surrogate function |

$\mathcal{S}$ | set of molecular species |

${S}_{i}$ | i-th molecular specie |

$\sigma $ | number of samples used to construct the surrogate |

t | time of the system |

$\mathbf{t}$ | vector of time points |

${t}_{k}$ | k-th time point |

$\tau $ | waiting time |

${\overrightarrow{v}}_{\mathrm{max}}$ | maximum velocity of the FST-PSO particles |

${\overrightarrow{v}}_{\mathrm{min}}$ | minimum velocity of the FST-PSO particles |

$\omega $ | inertia factor of FST-PSO |

${X}_{n}^{\mathbf{c}}\left({t}_{k}\right)$ | simulated amount of the species ${S}_{n}$ at time ${t}_{k}$ |

$\mathbf{x}\left(t\right)$ | vector representing the state of the system at time t |

${x}_{n}$ | amount of the n-th molecular specie |

$\chi $ | control point |

${\chi}_{Q}$ | Q-th control point |

$\xi $ | length of the individuals representing the DFs |

$\chi $ | vector of control points |

${y}_{\rho}$ | fitness value of the $\rho $-th point of the surrogate |

${z}_{\rho}$ | $\rho $-th point of the search space to build the surrogate function |

${\zeta}_{1}$ | random number sampled from an uniform distribution |

${\zeta}_{2}$ | random number sampled from an uniform distribution |

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**Figure 1.**Example of stochastic simulation: 100 independent SSA runs of the MM model—all starting from the same initial state ($S=200,E=100,ES=0$, and $P=0$) and using the same parameterization—lead to quantitatively different trajectories and a final distributions of the chemical species.

**Figure 3.**Effect of the anti-log DF. (

**Left**): control points encoding the DF. (

**Right**): effect of the mapping of a particle’s component (x axis) in the original search space of the PE problem (y axis).

**Figure 4.**(

**Left**): different examples of the linear transformation ${l}_{p}$ with different values of p. (

**Right**): examples of folding operators with varying values of p and r.

**Figure 5.**Graphical representation of the DF used in the PE problem. (

**Left**): anti-log function analytically designed to expand the fitness landscape in the lowest orders of magnitude. (

**Right**): evolved DF that substantially performs the same dilation of the anti-log DF.

**Figure 6.**Effect of DFs on the fitness landscape of the MM model. (

**a**) Original fitness landscape. (

**b**) Fitness landscape dilated by means of the analytical anti-log DF. (

**c**) Fitness landscape dilated by means of the optimal evolved DF.

**Figure 7.**(

**Left**): convergence plot of the PE of the MM model obtained with FST-PSO (no ${v}_{min}$) (coral solid line), FST-PSO using the analytic DF (green dashed line), and FST-PSO using the evolved DF (blue dotted line). The lines correspond to the ABF calculated over 30 runs. (

**Right**): boxplots representing the distributions of the best solutions found at the end of each run, for each methodology; red dashes denote the mean of the distributions, while diamonds denote the outliers. The asterisks denote the p-values obtained by comparing the distributions by means of the Mann–Whitney U tests ( ** p-value $\le 0.0001$).

**Figure 8.**Effect of the surrogate modeling on the fitness landscape of the MM model. (

**a**) Surrogate model of the original fitness landscape. (

**b**) Surrogate model of the fitness landscape dilated by means of the anti-log DF. (

**c**) Surrogate model of the fitness landscape dilated by means of the evolved DF.

**Figure 9.**Surrogate models of the fitness landscape for the PE of the MM model dilated using the evolved DF, with $\gamma =2$ (

**a**), $\gamma =3$ (

**b**), $\gamma =5$ (

**c**), $\gamma =10$ (

**d**), $\gamma =50$ (

**e**), $\gamma =100$ (

**f**).

**Figure 10.**(

**Left**): convergence plot of the PE of the MM model obtained with FST-PSO (no ${v}_{min}$) (coral solid line), FST-PSO using the analytic DF + surF (green dashed line), FST-PSO using the evolved DF + surF (blue dotted line). The lines correspond to the ABF calculated over 30 runs. (

**Right**): boxplots representing the distribution of the best solutions found at the end of each run; red dashes denote the mean of the distributions, while diamonds denote outliers. The asterisks denote the p-values obtained by comparing the distributions by means of the Mann–Whitney U tests (* p-value $\le 0.5$, ** p-value $\le 0.0001$).

Molecular Species | Amount |
---|---|

S (substrate) | 200 |

E (enzyme) | 100 |

$ES$ (enzyme–substrate complex) | 0 |

P (product) | 0 |

ID | Name | Semantics |
---|---|---|

0 | ${I}_{p}$ | Identity |

1 | ${l}_{p}$ | Linear transformation |

2 | ${\mathcal{F}}_{\frac{1}{4}}\left({l}_{p}\right)$ | |

3 | ${\mathcal{F}}_{\frac{1}{2}}\left({l}_{p}\right)$ | Folding operators |

4 | ${\mathcal{F}}_{\frac{3}{4}}\left({l}_{p}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Nobile, M.S.; Papetti, D.M.; Spolaor, S.; Cazzaniga, P.; Manzoni, L.
Shaping and Dilating the Fitness Landscape for Parameter Estimation in Stochastic Biochemical Models. *Appl. Sci.* **2022**, *12*, 6671.
https://doi.org/10.3390/app12136671

**AMA Style**

Nobile MS, Papetti DM, Spolaor S, Cazzaniga P, Manzoni L.
Shaping and Dilating the Fitness Landscape for Parameter Estimation in Stochastic Biochemical Models. *Applied Sciences*. 2022; 12(13):6671.
https://doi.org/10.3390/app12136671

**Chicago/Turabian Style**

Nobile, Marco S., Daniele M. Papetti, Simone Spolaor, Paolo Cazzaniga, and Luca Manzoni.
2022. "Shaping and Dilating the Fitness Landscape for Parameter Estimation in Stochastic Biochemical Models" *Applied Sciences* 12, no. 13: 6671.
https://doi.org/10.3390/app12136671