## 2 Tissue-like P Systems  
### TPs with Evolutional Symport/Antiport and Promoters  
As a variant of TPs, TPs with evolutional symport/antiport and promoters are based on the fact of biological reactions [62]. In classic TPs, substances or objects are transmitted through the execution of communication rules in different cells or regions, and communication between two given membranes is realized with the help of symport/antiport rules. In the presence of certain chemical substances, rules with promoters are executed to dynamically change the working manner of the computing model. The purpose of this ex tended P system is to modify and rewrite objects during the communication process using the framework of evolutionary mechanisms. Therefore, a recognizer TP with evolutional symport/antiport and promoters is designed and defined as a tuple in the following [9]:  
\[\Pi=(\Gamma,\varepsilon,\mu,\omega_{1},\cdots,\omega_{m},\mathrm{R},\sigma_{in},\sigma_{out})\]  
where  
1. \(\Gamma\) is nonempty finite set of alphabets consisting of elements, and all elements in this alphabet are called objects;
2. \(\varepsilon\) is a finite alphabet set of \(\Gamma\), which represents objects initially placed in the environment, such that \(\varepsilon\subseteq\Gamma\);
3. \(\mu\) is the structure of the P system, which consists of \(m\) membranes;
4. \(\omega_{1},\cdots,\omega_{m}\) are finite multisets of objects over \(\Gamma\), initially placed in \(m\) membranes, where \(\omega_{i}\subseteq\Gamma\), for \(1\leq i\leq m\);
5. R is a finite set of communication rules for the system, which contains two kinds of rules with promoters of the following restrictions. 1. Evolutional symport rules with promoters: \(\big{[}u\big{|}_{p}\big{]}_{i}\big{|}_{j}\rightarrow\big{[}v^{{}^{\prime}} \big{]}_{i}\big{[}u^{{}^{\prime}}\big{]}_{j}\), where \(0\leq i\
eq j\leq m\), \(u\in\Gamma^{+}\), \(u^{{}^{\prime}}\in\Gamma^{*}\), \(p\in\Gamma\), \(|u|>0\). It can only be applied to a configuration if both the multiset of objects \(u\) and promoter objects \(p\) appear in the same existing membrane \(i\). When such an evolutional symport rule with promoters associated with membrane \(i\) and membrane \(j\) is employed, the multiset of objects \(u\) under the presence of promoter objects \(p\) in membrane \(i\) is simultaneously sent to membrane \(j\) and evolved into new objects \(u^{{}^{\prime}}\);
2. Evolutional antiport rules with promoters: \(\big{[}u\big{|}_{p}\big{]}_{i}\big{[}v\big{]}_{j}\rightarrow\big{[}v^{{}^{ \prime}}\big{]}_{i}\big{[}u^{{}^{\prime}}\big{]}_{j}\), where \(0\leq i\
eq j\leq m\), \(u,v\in\Gamma^{+}\), \(u^{{}^{\prime}},v^{{}^{\prime}}\in\Gamma^{*}\),\(p\in\Gamma\), \(|u|,|v|>0\). It can only be applied to a configuration if both \(u\) and \(p\) appear in the same existing membrane \(i\), and another membrane \(j\) in the same configuration contains \(v\). When such an evolutional antiport rule with promoters associated with \(i\) and \(j\) is employed, \(u\) under the presence of \(p\) in \(i\) is simultaneously sent to \(j\) and evolved into \(u^{{}^{\prime}}\), at the same time, \(v\) in \(j\) is also simultaneously sent to \(i\) and evolved into new objects \(v^{{}^{\prime}}\);
3. \(\sigma_{in}\) represents an input membrane or region, where \(\sigma_{in}\in\{\sigma_{0},\sigma_{1},\cdots,\sigma_{m}\}\);
4. \(\sigma_{out}\) represents an output membrane or region, where \(\sigma_{out}\in\{\sigma_{0},\sigma_{1},\cdots,\sigma_{m}\}\). In the TPs with evolutional symport/antiport and promoters, unlike other objects involved in computation, the promoter objects do not directly take part in the execution process of evolutional symport/antiport rules. They are not modified or changed during computation. However, it only alters the order in which the evolutional symport/antiport rules are executed. In the presence of promoter objects, a greater number of corresponding evolutional symport/antiport rules are implemented within the fixation time.