Effective sampling trajectory optimisation for sensitivity analysis of biological systems

Abstract

Sensitivity analysis has been widely applied to study the biological systems, including metabolic networks, signalling pathways, and genetic circuits. The Morris method is a kind of screening sensitivity analysis approach, which can fast identify a few key factors from numerous biological parameters and inputs. The parameter or input space is randomly sampled to produce a very limited number of trajectories for the calculation of elementary effects. It is clear that the sampled trajectories are not enough to cover the whole uncertain space, which eventually causes unstable sensitivity measures. This paper presents a novel trajectory optimisation algorithm for the Morris‐based sensitivity calculation to ensure a good scan throughout the whole uncertain space. The paper demonstrates that this presented method gets more consistent sensitivity results through a benchmark example. The application to a previously published ordinary differential equation model of a cellular signalling network is presented. In detail, the parameter sensitivity analysis verifies the good agreement with data of the literatures.

1 Introduction

The dynamics of reaction networks in living organisms have been extensively studied in systems biology [1]. Simulation of biological and cellular processes is achieved through a range of mathematical modelling approaches [1]. However, the high complexity of the biological processes often results in the great uncertainty of model parameters and input states [2–4]. Use of these models relies on estimates of these kinetic parameters, which are often poorly constrained by the experimental observations [1, 3]. Sensitivity analysis can provide valuable insights about how robust the biological responses are with respect to the changes of kinetic parameters and which model inputs are the key factors that affect the model outputs [5]. Also, sensitivity analysis is valuable for guiding experimental analysis, model reduction, and parameter estimation [2, 5]. A number of approaches of sensitivity analysis of biochemical models have been developed. Local and global sensitivity analysis (GSA) approaches are the two types of sensitivity analysis that are commonly applied in systems biology. Local sensitivity analysis studies the impact of small and single perturbation on the model outputs. GSA has been applied to understand how the model outputs are affected by large and simultaneous variations of the model parameters [3, 6]. The GSA has a wider range of application [7, 8].

There are different GSA methods ranging from the screening method, the variance‐based methods to the distribution‐based methods. The Original Morris Method (OMM) plays an important role in the screening methods [9]. It is based on a few one‐at‐a‐time (OAT) runs. Only one factor is changed at a time so that the influence of the factor on the model output can be calculated [10–12]. The variance‐based methods quantify the input and output uncertainties as probability distributions, and decompose the output variance into parts attributable to input variables and their combinations [3, 13]. In the distribution‐based methods, the principle of sensitivity measures is to quantify the change in shape between the conditional and unconditional model output density function such as the distance metric for the difference in a probability distribution function (PDF), without reference on a particular moment [14, 15].

Global approaches are computationally demanding because they require many samples from the uncertain factor space (typically high‐dimensional) [1, 16]. Among the above‐mentioned methods, the Morris method has the minimum calculation burden and is particularly well suited for the case when the number of uncertain factors is large or the model is expensive to compute. However, since the limited number of sampling trajectories is not enough to cover the whole uncertain space, the Morris method is proved to be a compromise between accuracy and efficiency. Note that the trajectories refer to the special point sets of several sampling points from the uncertain space (see details in Section 3). The sensitivity results, based on the Morris method, often vary with the sampled trajectories.

To overcome this drawback, Campolongo et al. [3] put forward the Large Model Screening Method (denoted by LMSM), which selects a few Morris trajectories from numerous initial trajectories in such a way as to maximise their dispersion in the input space. This method may be unrealisable because it results in the ‘combinatorial explosion’ and needs a high computational demand. To solve this problem, J. Ribes et al. [17] applied an iterative procedure (denoted by MIF) to find the suboptimal Morris trajectories. Qiao [18] developed the quasi‐OTEE (quasi‐Optimised Trajectories based Elementary Effects) method (denoted by TS), which deletes the trajectories one by one until the number of the rest trajectories is required. Although these three methods, to some extent, can improve the coverage of sampling points, the analysis results are often inconsistent depending on the number of initial trajectories. Norton J. [19] argued that the sampling design based on maximising dispersion is not equal to generate a uniform design and even causes a poor scan. Furthermore, the time cost of the above three methods presents a geometric growth with the increasing number of initial trajectories.

Here, we propose a novel trajectory sampling design to improve the original Morris method. The sampling design combines the Bird Swarm Algorithm (BSA) with the Latin Hypercube Sampling (LHS) to get the uniform sampling, and the Morris method based on this sampling design is denoted by Trajectory‐Optimisation Morris method (TOM). The numerical example demonstrates that the novel sampling design can lead trajectories to effectively cover the whole uncertain space. Consequently, the sensitivity measures are accurate and stable.

The remainder of this paper is organised as follows. Section 2 introduces the governing ordinary differential equations (ODE) of the general biological models. Section 3 contains the detailed information on the trajectory‐optimisation Morris method and its numerical simulation results. In Section 4, the parameter sensitivity of a previously published ODE model involving cellular signalling is evaluated based on the TOM in contrast to the other methods. Section 5 makes a summary.

2 Governing ODEs of biological models

The complex biological pathways describing the cellular signalling and other metabolic phenomena consist of large numbers of simultaneous chemical reactions among many species. Consider a general model system of equations representing the biological pathways with M molecular species. Define $\mathit{c}(t)$ to be a vector of concentrations of each molecular species as a function of time $(c_1(t),c_2(t),\dots ,c_M(t))$ where $c_i(t)$ is the concentration of the i th species at time t [20]. The ODE model describing the chemical kinetics of this system is given by

$$\frac{d{c}_i(t)}{\mathrm{d}t}=g_i(\mathit{c}(t),\mathit{k}),i=1,\dots ,M$$ (1)

where $\mathit{k}=(k_1,k_2,\dots ,k_q)$ represents q kinetic parameters of the model. $\mathit{c}(0)=(c_1(0),c_2(0),\dots ,c_M(0))$ is the initial conditions.

We are interested in studying the sensitivity of some biological responses (e.g. transcription factor concentration) to the kinetic parameters or the initial conditions within their defined ranges. Take the model parameters as an example, each parameter $k_i$ is assumed to vary on an interval $[a_i,b_i]$ and the combination of all parameter intervals is defined to be the uncertain parameter space when the parameters are independent. Suppose the single model output, i.e. the interested biological response, can be written by

$$o(t)=h(\mathit{c}(t),\mathit{k})$$ (2)

The sensitivity of model output $o(t)$ with respect to any uncertain parameter or initial condition is a function of time since the output is time‐dependent. The sensitivities at different time points are usually concatenated into a sensitivity vector or are sum up as the general sensitivity index.

3 Trajectory‐optimisation Morris method

This section describes the detailed TOM algorithm starting with the original Morris method. Without loss of generalisation, to simplify the presentation, in this section, we suppose a general time‐independent model with m uncertain input factors as follows

$$y=f(x_1,x_2,\dots ,x_m)$$ (3)

where $x_1,x_2,\dots ,x_m$ is the uncertain input factors, and each of them is normalised to [0,1]. The uncertain factor space is a unit hyper‐cube.

3.1 Original Morris method

Morris discretised the factor space to an experimentation region of a regular m ‐dimensional p ‐level grid [3]. Each $x_i$ may take on values from $[0,1/(p-1),2/(p-1),\dots ,1]$ (usually taking p  = 4) [9, 11]. Define the Elementary Effect (EE) of the i th factors as

$$\begin{array}{rl}E{E}_i& =[f(x_1,\dots ,x_{i-1},x_i+\mathrm{\Delta },x_{i+1},\dots ,x_m)\\ & -f(x_1,\dots ,x_{i-1},x_i,x_{i+1},\dots ,x_m)]/\mathrm{\Delta }\end{array}$$ (4)

where step $\mathrm{\Delta }$ is a predetermined constant value, $\mathrm{\Delta }=p/2(p-1)$.

The input space is randomly sampled to get r starting points $s{p}_j(j=1,2,\dots ,r)$, and further Morris constructed r trajectories starting, respectively, from $s{p}_1,\dots ,s{p}_r$. The j th trajectory can be written as a matrix $B_j^{*}\in R^{(m+1)\times m}$. The any two adjacent rows of $B_j^{*}$, such as the i th row and the (i  + 1)th row, differ only in their i th elements.

$$B_j^{*}(i)=\left[\begin{array}{ccccccc}{x}_1^{*}& \dots & x_{i-1}^{*}& x_i^{*}& x_{i+1}^{*}& \dots & x_m^{*}\\ x_1^{*}& \dots & x_{i-1}^{*}& x_i^{*}+\mathrm{\Delta }& x_{i+1}^{*}& \dots & x_m^{*}\end{array}\right]$$ (5)

Through the r trajectories, the EEs of all input factors are calculated for r runs. For r runs (r trajectories), two sensitivity measures are defined as ${\mu }_i^{*}=\left(1/r\right){\sum }_{j=1}^r\left|\mathrm{E}{\mathrm{E}}_{i,j}\right|$ and ${\sigma }_i=\sqrt{\left(1/r\right){\sum }_{j=1}^r{(\mathrm{E}{\mathrm{E}}_{i,j}-{\mu }_i^{*})}^2},i=1,2,\dots ,m$, where $\mathrm{E}{\mathrm{E}}_{i,j}$ is EE of the i th factor in the j th run. ${\mu }_i^{*}$ is the measure of main effect of the i th factor on the model output, and ${\sigma }_i$ reflects on what extent $x_i$ interacts with other factors. In order to access ${\mu }^{*}$ and $\sigma$ simultaneously, some researches proposed a comprehensive sensitivity index $S_i=\sqrt{\mu {{}_i^{*}}^2+{\sigma }_i^2}$ [20]. Larger the sensitivity index of a factor is, this factor has a greater impact on the model output. The factors with the smaller sensitivity index can be set as its constant nominal value for the further model reduction [4].

3.2 Sampling trajectory optimisation

In the original Morris method, the values of starting points of trajectories are randomly taken on their grids. The step $\mathrm{\Delta }$ is predetermined and fixed for all trajectories. Moreover, the number of trajectories is set to be small (usually r  = 10) for the sake of the computation efficiency. It is obvious that the trajectory sampling strategy may lead to an improper coverage of the input space, which consequently causes the randomness of sensitivity results.

Some modified Morris method in the literature such as LMSM, MIF and TS, are only guaranteeing the dispersion of trajectories, and the sampling points are easy to fall onto the edge of the input space. This sampling design inevitably resulted in poor coverage and uniformity [19]. What's more, the larger the number of sampling trajectories is, the methods produce more combinations and need higher computation demand.

In this subsection, we aim at enhancing the trajectory design to allow for a better exploration of the input space. The LHS is used to generate the uniform starting points. An optimal step $\mathrm{\Delta }$ is found for each trajectory by an optimisation procedure to improve the uniformity of trajectories. Also, the modification of distance measurement improves the efficiency of computation.

3.2.1 Generating the starting points of trajectories by LHS

Latin hypercube sampling approach has been applied to many computer experiments since proposed by McKay et al. (1979) [21]. The LHS, a stratified sampling, can be applied to multiple variables and ensure that the distribution function of a factor can be sampled evenly. Compared with Monte–Carlo sampling, the LHS can reduce the number of runs and get sampling results with more accurate and reasonable allocation [22]. For an input space of m variables, the interval of each variable is first divided equally into n unit when the sampling size is n. This will divide the original hypercube into $n^m$ small hyper‐cubes. Let A be an n ‐by‐m matrix in which each column is a random permutation of series $\{1,2,\dots ,n\}$. Each row of A corresponds to a selected cube. Also, a sampling point is obtained by randomly selected in this small hypercube.

The LHS is called the spaced‐filled design because it can ensure that each unit is selected once in each interval. Fang pointed out that LHS, based on the uniformity criterion, could be considered as a uniform design [23, 24]. We plot the two‐dimensional sampling points both through random sampling and LHS in Fig. 1. As shown, the uniformity of the sampling points based on LHS is obviously better than the one based on the random sampling.

Fig. 1.

Comparison of 20 sampling points obtained by the LHS and random sampling

Here, the starting points of r trajectories are produced through the LHS. The interval of each input factor is equally divided into r ‐level grid instead of the p ‐level grid. The j th starting point can be written as $s{p}_j=(x_1^j,\dots ,x_m^j),j=1,2,\dots ,r$. The i th input factors $x_i^1,x_i^2,\dots ,x_i^r$ of r starting points randomly take values from its r ‐level grid to ensure every grid be sampled.

3.2.2 Modified distance measures between trajectories

The trajectory optimisation process needs to calculate the distance between two trajectories. Campolongo et al. [3] defined the distance between trajectory $j_1$ and $j_2$ as

$$d_{j_1{j}_2}=\left\{\begin{array}{cc}\sum _{a=1}^{m+1}\sum _{b=1}^{m+1}\sqrt{\sum _{i=1}^m{[{\mathit{x}}_a^{j_1}(i)-{\mathit{x}}_b^{j_2}(i)]}^2}& j_1\ne j_2\\ 0& \text{others}\end{array}\right.$$ (6)

where ${\mathit{x}}_a^{j_1}(i)$ indicates the i th factor value of the a th point of the j ₁ th Morris trajectory.

Apparently, the above distance calculation is complex and time‐consuming. To overcome this drawback, a modified distance calculation approach is proposed here as the dispersion index. Fig. 2 shows the geometry of the distance between trajectory $j_1$ and trajectory $j_2$ in the three‐dimensional input space. As shown, the j th trajectory corresponds to a hypercube with the side length ${\mathrm{\Delta }}_j$. Then, the Euclidean distance between the core points of two hyper‐cubes is taken as the distance measure between the two trajectories instead of (6). The modified distance measure can be written as

$${\tilde{d}}_{j_1{j}_2}=\left\{\begin{array}{cc}\sqrt{\sum _{i=1}^m{[c{o}_{j_1}(i)-c{o}_{j_2}(i)]}^2}& j_1\ne j_2\\ 0& \text{others}\end{array}\right.$$ (7)

where $c{o}_{j_1}(i)$ indicates the i th coordinate of the core point of the j ₁ th Morris trajectory.

Fig. 2.

Geometric representation of the new distance between the trajectory

$j_1$

and the trajectory

$j_2$

in the three‐dimensional space

Due to the good uniformity of starting point distribution, this modified distance measure is reasonable. Moreover, the core point of a hypercube is the midpoint of the connection line between the initial point and the end point of the related trajectory. The modified distance measure is easy to compute and a great deal of computing time is reduced.

3.2.3 Finding the optimal steps by BSA

The Morris method has three important elements: starting points, step, and the construction rule of trajectories. The original Morris method uses a constant step $\mathrm{\Delta }$ to disturb the value of each input factor, which limits, to some extent, the span of all sampling points. An important advantage of using variable steps is that the sampling points have a wider span range with a better uniform distribution.

For the j th trajectory, the step ${\mathrm{\Delta }}_j$ is redefined as

$${\mathrm{\Delta }}_j=l_j\times v_j,j=1,2,\dots ,r$$ (8)

where $l_j$ is the shortest distance between the j th starting point and the other starting points. $v_j$ is a scalar coefficient between 0 to 1. Using the distance $l_j$ as the limit of ${\mathrm{\Delta }}_j$ can avoid the j th trajectory pushing into the hypercube space of the other trajectories.

The maximum distance between r trajectories is denoted by

$$d_{\text{max}}=\text{max}({\tilde{d}}_{j_1{j}_2}),j_1\ne j_2\text{and}{j}_1,j_2=1,\dots ,r$$ (9)

The minimum distance between r trajectories is denoted by

$$d_{\text{min}}=\text{min}({\tilde{d}}_{j_1{j}_2}),j_1\ne j_2\text{and}{j}_1,j_2=1,\dots ,r$$ (10)

The maximin and minimax principles are the basic guideline of uniform design [25]. Minimising the maximum distance $d_{\text{max}}$ is to ensure the compactness of the trajectories and prevent sampling points from running out of the input space. Maximising the minimum distance $d_{\text{min}}$ is to ensure the dispersion of sampling trajectories. According to the maximin and minimax principle, the optimal scalar coefficient vector $\mathit{v}=[v_1,v_2,\dots ,v_r]{}^T$ could be found by the optimation problem

$$\left\{\begin{array}{rl}& \text{min}\mathit{J}=\text{min}(d_{\text{max}}/d_{\text{min}})\\ & s.t.v_j\in (0,1],j=1,2,\dots ,r\end{array}\right.$$ (11)

The non‐linear problem can be solved by using a different intelligent optimisation algorithm. The bird swarm optimisation algorithm is an effective bio‐inspired algorithm, which is based on the swarm intelligence extracted from the social behaviours: foraging behaviour, vigilance behaviour and flight behaviour. The position and role (producer and scrounger) of a bird is updated based on the models of these three behaviours. In contrast to the Particle Swarm Optimisation (PSO) and the Genetic Algorithm (GA), the BSA algorithm can significantly improve the search ability and convergence speed with a good diversity, which will reduce the calculating cost and get the accurate analysis results [26, 27]. Here, we look for the optimal‐scale coefficient vector v by the BSA. Once the v is determined, we can get the ${\mathrm{\Delta }}_i$ by (8) and consequently construct the sampling matrix to calculate sensitivity measures.

3.3 Proposed TOM algorithm flow

Here, we give the details of the Trajectory‐Optimisation Morris method (TOM) based on the optimal trajectory design presented above. This sensitivity measure process consists of four parts: (i) Generate starting points by the LHS. (ii) Optimise the steps by the BSA, based on a modified distance measure and the maximin and minimax principle. (iii) Produce optimal trajectories (sampling matrixes). (iv) Calculate the sensitivity measures. The specific flow diagram is as shown in Fig. 3. The VM is the population initialisation matrix about $v_j$.

Fig. 3.

Specific flow diagram of the proposed TOM algorithm

3.4 Numerical result

A benchmark function, named g ‐function [28], is utilised to evaluate the TOM method. The simulation results are compared with the results by the OMM, LMSM, MIF, and TS.

The g ‐function with m input variables is defined as

$$y=\prod _{i=1}^m{g}_i(x_i),g_i(x_i)=\frac{|4x_i-2|+a_i}{1+a_i}$$ (12)

where $a_i\ge 0$ and $x_i$ follows the uniform distribution of [0,1].

The importance of input factor $x_i$ is dependent on the value of $a_i(i=1,2,\dots ,m)$. The smaller $a_i$, the more important $x_i$ is. Once the values of all $a_i$ are determined, the relative importance of all input factors are determined [28, 29]. Here, we consider a g ‐function with 12 input factors and set the value of $a_i$ as shown in Table 1. From Table 1, we can theoretically determine the sensitivity ranks of the 12 input factors.

Table 1.

Values of $a_i$ in g ‐function

$a_i$	$a_1$	$a_2$	$a_3$	$a_4$	$a_5$	$a_6$
value	0.93	5.25	5.54	1.23	0.78	1.26

$a_i$	$a_7$	$a_8$	$a_9$	$a_{10}$	$a_{11}$	$a_{12}$
value	0.04	0.79	2.35	12.3	4.32	9.21

There is a consensus that if the sampling points are enough even to cover the entire input space, the sensitivity analysis results of the sampling‐based approaches will be accurate and stable. The original Morris method is a compromise between accuracy and efficiency through randomly sampling the input space with the limited number of trajectories. Through enhancing the trajectory design, the presented TOM method aims to get more accurate and stable results than the original method. The sensitivity measures ${\mu }^{*}$ and $\sigma$ are computed on the g ‐function by the TOM as well as OMM, LMSM, MIF, and TS for contrast. The simulation by each method is repeated 10 times at r   =  10. Figs. 4, 5, 6, 7–8 shows the simulation results. Due to the heavy calculation load of the LMSM, it is assumed that 10 trajectories are selected only from 30 random trajectories (10 out of 30). Totally, 10 trajectories are selected from 1000 random trajectories (10 out of 1000) for MIF and TS. Compared to OMM, each improved method including the LMSM, MIF, TS, and TOM calculates more consistent sensitivity results for 10 times simulations. Furthermore, it can be found out that the sensitivity measures by the TOM have the best consistency and smallest randomness with accurate sensitivity values.

Fig. 4.

Analysis results of the OMM method for 10 experiments

Fig. 5.

Analysis results of the LMSM method (10 out of 30) for 10 experiments

Fig. 6.

Analysis results of the MIF method (10 out of 1000) for 10 experiments

Fig. 7.

Analysis results of the TS method (10 out of 1000) for 10 experiments

Fig. 8.

Analysis results of the presented TOM method for 10 experiments

Table 2 shows the $S_i$ index ranks in one experiment and total computational time for 10 experiments (obtained with MATLAB using a PC with a 2.3 GHz Intel Core i5 processor). The BSA optimisation process is executed 200 iterations (as same as the next simulations). Note that the experiments only suppose a relatively small number of initial sampling points in LMSM, MIF, and TS, i.e. LMSM for 30, MIF and TS for 1000. When the number of initial points increases, the consuming time will rise greatly. The TOM reduces calculation time a lot compared to the LMSM. Besides, according to the theoretical ranks, the TOM has the more accurate and consistent sensitivity results than the OMM, LMSM, MIF, and TS. The difference between the TOM's and theoretical ranks lies on the input factors whose sensitivity measures are very close. Although much more than the OMM, the computing time of the TOM is comparative to the MIF and TS. This is because the TOM adopts the modified distance measures between trajectories and a simple and optimal trajectory design method. Yet the TS, MIF, and LMSM need to randomly generate numerous initial trajectories and then select a few trajectories according to some strategies, which inevitably cause ‘combination explosion’ and the selection process is time‐consuming.

Table 2.

Comparison of S index, the rank of an experiment and total time consuming of five methods for 10 experiments

Method		S index & Ranks												Time consuming
OMM	S index	9.40	7.60	6.75	6.58	4.75	4.24	3.77	2.63	2.34	2.31	0.74	0.70	0.52 s
	ranks	$x_7$	$x_5$	$x_6$	$x_1$	$x_8$	$x_4$	$x_9$	$x_2$	$x_3$	$x_{11}$	$x_{10}$	$x_{12}$
LMSM	S index	9.03	7.74	6.16	4.77	4.39	4.06	3.46	2.09	1.92	1.70	1.10	0.77	31 min
	ranks	$x_7$	$x_8$	$x_5$	$x_1$	$x_6$	$x_4$	$x_9$	$x_2$	$x_3$	$x_{11}$	$x_{12}$	$x_{10}$
MIF	S index	11.95	9.93	9.36	7.97	6.57	6.31	3.68	3.41	3.06	2.22	2.11	1.47	145.1 s
	ranks	$x_7$	$x_5$	$x_8$	$x_1$	$x_6$	$x_4$	$x_9$	$x_{11}$	$x_3$	$x_2$	$x_{12}$	$x_{10}$
TS	S index	11.13	9.50	8.76	8.44	7.20	6.50	3.44	3.36	3.01	2.63	1.23	1.08	140.7 s
	ranks	$x_7$	$x_5$	$x_8$	$x_1$	$x_6$	$x_4$	$x_9$	$x_{11}$	$x_2$	$x_3$	$x_{10}$	$x_{12}$
TOM	S index	5.76	4.31	3.84	3.59	3.00	2.76	2.46	1.91	0.98	0.95	0.87	0.74	121.6 s
	ranks	$x_7$	$x_5$	$x_8$	$x_1$	$x_6$	$x_4$	$x_9$	$x_{11}$	$x_3$	$x_2$	$x_{12}$	$x_{10}$
Theoretical ranks	Ranks	$x_7$	$x_5$	$x_8$	$x_1$	$x_4$	$x_6$	$x_9$	$x_{11}$	$x_2$	$x_3$	$x_{12}$	$x_{10}$

Furthermore, the difference of the computing time among the four concerned methods will become more and more obvious with the number of the input factors increasing. The time cost of four algorithms consists of two parts: the time of obtaining sampling trajectories and the time of running the models to get the sensitivity measures. When the number m of input factors increases, the time of the second part will grow equally for all algorithms. Thus, the difference lies on the first part — obtaining sampling trajectories. The time complexity of distance calculation is, respectively, $O(m)$ for the TOM method and $O(m^3)$ for the other concerned method according to (6) and (7). Therefore, the time cost of distance calculation for the LMSM, MIF, and the TS will grow rapidly with the number of input factors increasing, while the time for the TOM is much smaller. On the other hand, with the number of input factors increasing, the TS, MIF, and LMSM have to generate more initial trajectories, which directly causes that the time of the selection process for the sampling trajectories presents a geometric growth.

In statistics or mathematical analysis, the $\mathrm{C}{\mathrm{L}}_2$ index or $\mathrm{D}{\mathrm{L}}_2$ index is a metric to evaluate the uniformity of sample points in a space. Compared to $\mathrm{D}{\mathrm{L}}_2$ index, the $\mathrm{C}{\mathrm{L}}_2$ index is based on the centred point to judge the uniformity of sampling points and easy to calculate [30]. Here, the $\mathrm{C}{\mathrm{L}}_2$ index is used to measure the overall uniformity of the sampling points generated by the trajectory optimisation design. The $\mathrm{C}{\mathrm{L}}_2$ index for $n\times m$ sampling points is calculated as [30]

$$\begin{array}{rl}\mathrm{C}{\mathrm{L}}_2(Q){}^2& ={\left(\frac{13}{12}\right)}^2-\frac{2}{n}\sum _{s=1}^n\prod _{i=1}^m\left(1+\frac{1}{2}\left|x_{si}-\frac{1}{2}\right|\right.\\ & -\left.\frac{1}{2}{\left|x_{si}-\frac{1}{2}\right|}^2\right)+\frac{1}{n^2}\sum _{s=1}^n\sum _{t=1}^n\prod _{i=1}^m\left(1\right.\\ & +\frac{1}{2}\left|x_{si}-\frac{1}{2}\right|-\frac{1}{2}{\left|x_{ti}-\frac{1}{2}\right|}^2\\ & -\left.\frac{1}{2}|x_{si}-x_{ti}|\right)\end{array}$$ (13)

where Q is the set of sampling points ${\mathit{x}}_s=(x_{s1},x_{s2},\dots ,x_{sm}),s=1,\dots ,N$. When $\mathrm{C}{\mathrm{L}}_2$ index is used for the trajectory evaluation, N equals to $r(m+1)$ and represents the number of total sampling points of r trajectories.

A smaller mean or standard deviation of the $\mathrm{C}{\mathrm{L}}_2$ index indicates that the sample points show the more uniform distribution [24, 30]. Fig. 9 shows the mean and standard deviation of the $\mathrm{C}{\mathrm{L}}_2$ index for the OMM, MIF, TS and TOM while r  = 5,10,20,30,50. Regardless of the values of r, the mean and standard deviation of the $\mathrm{C}{\mathrm{L}}_2$ index by the TOM is the smallest and tends to be constant. It can be concluded that the enhanced trajectory design achieves a good uniform distribution even if the size of sampling points is small. Therefore, we can infer that sensitivity measures by the TOM are stable, and the sensitivity analysis result changes little when r is set to some different values. Fig. 10 shows the S indexes of 10 repeated experiments by the TOM while r  = 5,15,20,30, respectively. It is clear that our inference is the fact as shown in this figure. The TOM method is less sensitive to the value of r. The impact of r on analysis results becomes small. The TOM based on the optimal trajectory design can calculate stable sensitivity measures with a small sample size, which is suitable for the case when the number of input factors is high or the model is expensive to compute.

Fig. 9.

$\mathrm{C}{\mathrm{L}}_2$

index of sampling points based on four methods

Fig. 10.

S index at r = 5, 15, 20, 30 by the TOM method for 10 experiments

In addition, it should be noted that $\mathrm{C}{\mathrm{L}}_2$ indexes by the MIF and TS is larger than the indexes by the original Morris method. It is likely due to the fact that MIF and TS select specific trajectories through maximising the distances of trajectories. As previously described, most of the selected sampling points may consequently fall close to the edge of the input space, whereas few points are in the inner input space when the r is big. In other words, only maximising distance is not equal to generate a uniform design and may cause a poor coverage of the input space [19].

In conclusion, the TOM method presented here achieves three improvements: (i) Uniform distribution of sampling points through the optimal trajectory design; (ii) Stable and accurate sensitivity measures even if a small sample size; (iii) Reduced computation time which is suitable for the high dimensional and complex models.

4 Application to a cellular signalling network

In this section, we demonstrate the ability of the TOM to study various sensitivity measures and the sensitivity dynamics of a cellular signalling network. As a model problem, we use the mathematical model of the crosstalk between Mitogen‐activated Protein Kinases (MAPK) pathway and Smad‐dependent TGF‐β (Transforming Growth Factor $\beta$) signal transduction pathway from [31]. In [31], the detailed mathematical model on the crosstalk integrates two individual signal pathways and explains two conflicting performances in the experiments. However, only the crosstalk parameters are analysed through the ordinary Morris method to demonstrate two competing effects of MAPK cascade activation depending on the specific cell types. We then go on to study the crosstalk model in greater detail with the proposed TOM algorithm. We conclude that stable and reliable sensitivity measures for all uncertain parameters can be obtained and a few key parameters are picked out by the TOM. Further, the biological mechanism is illustrated through the time‐dependent sensitivity dynamics.

4.1 Model description

The research of mathematical models on signal transduction pathways has a great promotion significance not only on the research of life sciences but also on the treatment of related diseases [32]. TGF‐β family of proteins is involved in regulating a variety of cellular processes such as cell proliferation, apoptosis, and tumour invasion/metastasis [33]. TGF‐β mainly activates the Smad protein signalling. Meanwhile TGF‐β, as well as other extracellular stimuli (cytokines, epidermal growth factor, etc.), also induces the activation of MAPK cascade. Broad evidence exists for the crosstalk between the MAPK pathway and Smad‐dependent TGF‐β signal transduction pathway. Although other works have focused on the crosstalk, there are no mathematical model and consensus mechanisms description for the crosstalk. In [31], a detailed mathematical model on the crosstalk was developed through integrating the MAPK pathway with TGF‐β /Smad pathway.

The crosstalk model involves 24 cellular molecular species and 22 uncertain parameters. The governing ODEs written as (1) can be found in the literature [31]. The single model output represents the concentration of nuclear phosphorylated Smad complex, i.e. transcription factor, which determines cell proliferation or apoptosis. The nominal values and biological interpretations of all uncertain parameters as well as the initial conditions of all molecular species can be found in the literature [31–33]. A continuous 2 ng/ml TGF‐β stimulation is added for the simulation.

The uncertain parameter set consists of the parameters of TGF‐β /Smad pathway, the parameters of MAPK pathway and the parameters involved in the crosstalk. It should be noted that the crosstalk is highly cell‐type dependent. Parameter sensitivity analysis provides a powerful tool to identify the contribution of individual factors to the signalling response no matter what the cell type is. We investigated the influence of the uncertain parameters on model output, i.e. transcription factor concentration. The proposed sensitivity measure process using the TOM is executed as well as the original method (OMM), MIF and TS for contrast.

4.2 Results and discussion

For the parameters describing the TGF‐β /Smad pathway, the uncertain ranges are set to 50–150% of the nominal values. The uncertain ranges of the other factors involving the crosstalk are set to be much larger to take account into wider cell types [31]. The sensitivity measures with respect to the 22 parameters are time functions. The sum of sensitivity values at different time points is taken as the general sensitivity index. To evaluate the consistency of sensitivity measures, the simulation is repeated five times, respectively, using the OMM, MIF, TS, and TOM. Fig. 11 shows the general sensitivity index by four methods while the number of trajectories r is set to 10. It is clear that the sensitivity measures by the OMM is with maximum randomness and TOM is the best one with more consistent measures. For example, the parameters 3,4,5 from OMM and MIF are often near to the lowest grade of importance, while they keep in the much higher grade of importance from TS and TOM. TOM computes a more stable sensitivity measures than OMM and TS for the parameters 19–22 describing the crosstalk. The information shown in Fig. 11 d convinces us the importance grade of 22 uncertain parameters for guiding biological experimental design and analysis. Table 3 shows the time‐consumption for the OMM, MIF, TS, and TOM in an experiment. It demonstrates that the TOM, compared to the MIF and TS, can also get analysis result for the actual biological model with less time.

Fig. 11.

Sensitivity index of 22 uncertain parameters in the crosstalk model respectively by

(a) OMM, (b) MIF, (c) TS, (d) TOM

Table 3.

Comparison of the time consumption of 4 methods for a simulation

Method	OMM	MIF	TS	TOM
time consumption	67.09s	128.76s	128.40s	106.15s

The mean and standard deviation of EEs for each parameter are plotted in Fig. 12 when the TGF‐β /Smad pathway is with and without crosstalk, respectively. The analysis is based on the TOM (as same as the next simulations). As shown, EEs of the parameters 1,3,4,5,16, and 18 have the higher mean and standard deviation, and they are separated far from the cluster of other parameters. They are the first six important parameters for both of the two models which is vital for the parameter estimation. We can set the other parameters as their nominal values for the further model reduction. It is interesting that there is no difference in the key parameters selected from the two models. This is because that the Smad‐dependent TGF‐β signal transduction still occupies a dominant position in the signal transduction. The MAPK pathway has a relatively small impact, which agrees well with the experimental data [33, 34].

Fig. 12.

Sensitivity measure plot based on

(a) The crosstalk model and, (b) The original model with no crosstalk

To have the further insight into the signal transduction process, the time‐dependent sensitivity profiles of the output with respect to the six key parameters are plotted in Fig. 13 a. As a whole, the parameter 18, dissociation rate of nuclear Smad complex, has the greatest effect on the transcription factor concentration. Its sensitivity value peaks at ∼65 min after TGF‐β addition and then declines to the basal level after about 4–5 h which agrees well with the experimental data observed by Zi et al. [5]. The parameters 1,3,4, describing the ligand binding and transfer, begin to play significant roles after the downstream reactant, phosphorylated R‐Smad tends to saturation. The sensitivity value of parameter 16 representing the formation rate of Smad complex reaches to peak and declines to constant earlier than parameter 18, which agrees that the dissociation of nuclear Smad complex is followed by the formation of cytoplasmic Smad complex. The dynamic characteristics of the sensitivity profiles highly illustrate the biological mechanism [32–34].

Fig. 13.

Time‐dependent sensitivity profiles of the output with respect to

(a) 6 key parameters and (b) 4 interaction parameters

With respect to the crosstalk, two parameters $V_0$, $\gamma$, respectively, determine the intensity and duration of the crosstalk. The other two parameters $K_1$ or $K_3$, respectively, describe the inhibition and enhancement effects. The sensitivity profiles of crosstalk parameters are shown in Fig. 13 b, which have consistent dynamics to the profiles in [31].

In order to look further into the sensitivity analysis, we plot the time‐dependent $\left|\text{EE}\right|$ (the absolute EE) profiles of parameter 18 for 10 runs in an experiment of the TOM, shown in Fig. 14. It is clear that the EEs are distributed uniformly, which demonstrates the sampling points of the parameter 18 are enough scattered in its interval. Whereas in a same experiment of the OMM, we find out that the EE profiles are very close for 10 runs (the figure omitted). We assume that the sampling trajectories by the OMM, MIF, and TS may concentrate in a small area due to a poor scan of the input space and consequently cause the very closed absolute EEs values.

Fig. 14.

Time‐dependent

$\left|\text{EE}\right|$

profiles of parameter 18 with respect to 10 runs

From the above results, we can conclude that Smad‐dependent signal transduction still occupies a dominant position in the crosstalk, whereas the MAPK pathway has a relatively small impact. Only six key parameters out of 22 are selected for the further model reduction and parameter estimation. It can be concluded that the sensitivity analysis results by the TOM algorithm are reasonable but are more stable than other methods, which demonstrates the effectiveness of the TOM algorithm.

5 Conclusion

Although the traditional Morris method is fast and easy to implement, the analysis results are highly unstable because its trajectory design causes the poor coverage of the uncertain input space. Here, a novel trajectory‐optimisation algorithm is developed to improve the Morris method through a better space scan. The proposed TOM method improves the uniformity of the starting points of trajectories by the LHS, implements the uniform design of sampling trajectories by bird swarm optimisation algorithm and reduces the computation burden by the modified trajectory distance measure. The TOM method can achieve stable sensitivity measure even for a small sample size and is suitable for high‐dimensional complex biological models.

The application of the TOM method to the model of the crosstalk between MAPK pathway and TGF‐β /Smad pathway is discussed in detail here. Six key parameters out of 22 are selected for the further model reduction and parameter estimation. The sensitivity measures of all parameters show that the Smad‐dependent signal transduction still occupies a dominant position in the crosstalk, whereas the MAPK pathway has a relatively small impact. The dynamic characteristics of the sensitivity profiles highly illustrate the biological mechanism. The data obtained from the analysis of the sensitivity profiles agree well with the experimental data in the literature. Since the TOM method can achieve a stable and reliable sensitivity measure with a low computation burden, we can treat the TOM method as an effective GSA tool to provide useful information on biological systems.