Research Article | Open Access
Bacterial Colony Optimization
This paper investigates the behaviors at different developmental stages in Escherichia coli (E. coli) lifecycle and developing a new biologically inspired optimization algorithm named bacterial colony optimization (BCO). BCO is based on a lifecycle model that simulates some typical behaviors of E. coli bacteria during their whole lifecycle, including chemotaxis, communication, elimination, reproduction, and migration. A newly created chemotaxis strategy combined with communication mechanism is developed to simplify the bacterial optimization, which is spread over the whole optimization process. However, the other behaviors such as elimination, reproduction, and migration are implemented only when the given conditions are satisfied. Two types of interactive communication schemas: individuals exchange schema and group exchange schema are designed to improve the optimization efficiency. In the simulation studies, a set of 12 benchmark functions belonging to three classes (unimodal, multimodal, and rotated problems) are performed, and the performances of the proposed algorithms are compared with five recent evolutionary algorithms to demonstrate the superiority of BCO.
Swarm intelligence is the emergent collective intelligent behaviors from a large number of autonomous individuals. It provides an alternative way to design novel intelligent algorithms to solve complex real-world problems. Different from conventional computing paradigms [1–3], such algorithms have no constraints of central control, and the searching result of the group will not be affected by individual failures. What is more, swarm intelligent algorithms maintain a population of potential solutions to a problem instead of only one solution.
Nowadays, most of swarm intelligent optimization algorithms are inspired by the behavior of animals with higher complexity. Particle swarm optimization (PSO) [4, 5] was gleaned ideas from swarm behavior of bird flocking or fish schooling. Ant colony optimization (ACO) was motivated from the foraging behavior of ants [6, 7]. Artificial fish swarm algorithm (AFSA) was originated in the swarming behavior of fish , and artificial bee colony algorithm (ABCA) [9, 10] was stimulated by social specialization behavior of bees. However, the states of the abovementioned animals are more complex, and their behaviors are difficult to describe qualitatively.
As prokaryote, bacteria behave in a simple pattern which can be easily described. Inspired by the foraging behavior of Escherichia coli (E. coli) in human intestines, Passion proposed an optimization algorithm known as bacterial foraging optimization (BFO) recently . In the same year, another well-known study based on bacterial behavior, bacteria chemotaxis (BC), was presented by Müller et al. . Those two papers broadened the bacterial optimization area with a new horizon. Since then, a growing number of researchers who paid great attention to this field began to divert their concentrations on this new algorithm [13–15] and extended the concept to algorithms such as fast bacterial swarming algorithm (FBSA) , bacterial-GA foraging algorithm , and adaptive bacterial foraging algorithm (ABFO) . Except those new proposed algorithms, improvements of bacterial foraging optimization (BFO) were considered by investigators. Some of the key models involve the discussion of chemotaxis step length (BFO-LDC and BFO-NDC ) or adaptive chemotaxis step . These researches on bacterial foraging optimization suggested that predicting and controlling the dynamical behavior of microbial colonies might have profound implications in reality.
However, traditional bacterial behavior-based algorithms (BFO or BC) only considered individual behaviors instead of social behaviors with swarm intelligence. Each individual in the colony independently searches for food by their own experience without any information exchange with others. What made the situation worse is the complicated characteristics of the original bacterial behavior-based algorithms. Taking BFO for example, long period of time has been spent on random chemotaxis. Additionally, the chemotaxis, reproduction, elimination and dispersal processes are inner iterations that lead to high computation complexity. The frequency of chemotaxis, reproduction, elimination, and dispersal is predefined without considering the environmental dynamics.
To deal with the aforementioned problems, we propose a new bacterial behavior based optimization algorithm—bacterial colony optimization (BCO)—to formulate bacteria's behavior with swarm intelligence. The main contributions of this paper are described as follows.(i)A new description of artificial bacteria lifecycle is formulated, which include five basic behaviors and their corresponding models.(ii)Newly created bacterial behavior model is proposed to simplify the bacteria optimization process.(iii)A novel chemotaxis and communication mechanism is used as well to update the bacterium positions.(iv)Two methods of communication: individual interaction and group exchange are introduced to improve the optimization efficiency.
The rest of the paper is organized as follows. Section 2 describes the basic behavior of artificial bacteria, and their corresponding models and the proposed algorithm are proposed in Section 3. Section 4 presents the results of the simulation studies, followed by the conclusion in Section 5.
2. Artificial Bacteria Behavior
Bacteria swim by rotating whip-like flagella driven by a reversible motor embedded in the cell wall (Figure 1(a)). As environment changed, the fittest individuals will survive while the poorer ones will be eliminate. After a while, the existing bacteria generate new offsprings (Figure 1(b)). With the depletion of nutrition and increasing of population, the resources can no longer hold the bacteria population. Some individuals may choose to migrate to a new area with better nutrition supply (Figure 1(c)). During the searching for a new habitat, information sharing and self-experience are both essential (Figure 1(c)). This process brings together the macro- and the microscale that allows bacteria to make chemotaxis movements as well as interacting with other bacterium. This can be realized by incorporating the mechanism of chemotaxis and communication into the whole optimization process. Apart from the mechanism of chemotaxis and communication, three additional mechanisms, reproduction, elimination, and migration are also viewed as evolutionary operators to cover the entire optimization search process (Figure 1(d)).
The basic behavior of bacteria in the lifecycle can be simply divided into five parts: chemotaxis, elimination, reproduction, migration, and communication. The detail descriptions of those processes are given as follows.
A fascinating property of E. coli is their chemotactic behavior. Flagellated bacteria E. coli produce motion by the movement of their flagellum. The whole process of chemotaxis can be depicted into two operations: running and tumbling. In the process of running, all flagella rotate counterclockwise to form a compact propelling the cell along a helical trajectory. In this way, bacteria can swim straightly in one direction. In the case of tumbling, the flagella rotate clockwise, which pull the bacterium in different directions .
Bacteria need to migrate up to the concentration gradient of nutrients. Hence, the alteration between two operations in chemotaxis must be well organized. A basic strategy used by microbes is to move in one direction for several steps. If the new environment cannot satisfy the bacteria, then they would tumble to pull themselves into a new direction and start a second run.
2.2. Elimination, Reproduction and Migration
Based on the theory of nature selection, bacteria with poorer searching ability would have higher chance of being eliminated. In contrast, those who perform well in the chemotaxis process would obtain more energy for survival and thus have a high probability of reproduction. In our proposed model—bacteria colony optimization (BCO), artificial bacteria with high quality in searching for nutrition have the opportunity to be endowed a relevant level of energy grade. Whether a bacterium has the chance to reproduce or not would base on the level of its energy grade.
After a long time of chemotaxis, elimination, and reproduction in the same area or surroundings, the nutrition must be used up or cannot satisfy all the bacteria. At this time, some bacteria have to migrate into a new nutritious place, and this process is called “Migration.”
Communication is an essential behavior that exists in whole processes of bacterial life. Three basic communication mechanisms are employed in the bacteria colony optimization (see Figure 2), including dynamic neighbor oriented (Figure 2(a)), random oriented (Figure 2(b)) and group oriented (Figure 2(c)), which represent three types of topologic structures. The bacteria share information between individuals in the first two exchange mechanisms, and the third mechanism (group oriented) means that bacteria communicate searching information in unit if groups.
(a) Dynamic neighbor oriented
(b) Random oriented
(c) Group oriented
3. Bacterial Colony Optimization Principle
3.1. Lifecycle Model
The behavior of artificial bacteria in this paper includes five parts, but those behaviors are continuous, mingle, and amalgamate. Chemotaxis behavior is always accompanied by communication along the whole lifecycle. Therefore, chemotaxis and communication are treated as one model in Bacterial Colony Optimization (BCO). Bacteria have two chooses after long times of chemotaxis and communication. They may die for the lack of food, or they may reproduce if they are capable of searching for food. Within the complicated environment, some individuals may run into dangerous place (go out of boundary). Specific situations like this also worth special treatment in lifecycle model (LCM). Migration conducts as an independent model, which involves energy depletion, group diversity, and chemotaxis efficiency. The overall model of bacterial lifecycle is shown in Figure 3.
The framework of this model is based on an agent-environment-rule (AER) schema; that is, there are three fundamental elements: agent, environment, and rule. The detailed description is listed below:(i)A: artificial bacteria,(ii)E: artificial environment,(iii)R: the environment/organism interaction mechanisms.
LCM model is different from the original population-based model in which all the individuals share the same state properties. LCM is a philosophy that embraces the uniqueness of the individuals in a system with multiple individuals that have its own set of state variables and parameters. Looking in state space, the population is akin to clouds of individuals with similar behaviors, and other clouds, amounting to separate individuals. Fundamentally, this allows for individuals to exist and speciation to occur and permits extinction. In general, the lifecycle model of artificial bacteria in BCO can be divided into four submodels: chemotaxis and communication model, reproduction model, elimination model, and migration model.
The detailed explanations of each submodel are formulated in the following.
3.2. Chemotaxis and Communication Model
Chemotaxis is accompanied with communication in the whole optimization process. Bacteria run and tumble in the competitive environment. However, they also have to offer their information to the colony in exchange of overall information which would guide them in direction and ways of movement. As is shown in Figure 4, bacterial chemotaxis is directed by three elements: group information, personal previous information, and a random direction. All three factors conduct bacteria running and tumbling toward optimum.
Bacterium runs or tumbles with communication process can be formulated as:
Actually, the above position updating equations only consider the relationship between the individuals and the group. The bacteria share information between individuals also merged into the communication model. Pseudocode for E. coli chemotaxis and communication is listed in Pseudocode 1.
3.3. Elimination and Reproduction Model
Each bacterium is marked with an energy degree based on its search capability. The higher level of energy indicates a better performance of bacterium. The level of energy decides the probability of elimination and reproduction. The distribution of bacterial energy degree was sorted and analyzed and then used as a criterion to judge the qualification of the bacteria. The details are summarized as
All behaviors of bacteria were restricted within a restrained area. As a general principle, individuals are not allowed to go out of the region, so boundary control is especially important. If bacteria move away from the feasible domain, at least two strategies will be performed based on experiences. One is to generate new individuals to replace the outer ones, and the other is to let the outer ones stay at boundary but change the forward direction to keep them effectiveness. In this paper, those outer individuals named “unhealthy” are put into a set which hold the candidate bacteria for elimination.
3.4. Migration Model
Naturally, bacteria could pursue more nutrition by migration. In optimization aspect, migration can avoid local optimum within some distance. Especially, the migration of artificial bacteria in BCO is not based on a certain given probability. It depends on a given condition. When condition is fulfilled, bacterium would migrate to a new random place, as described by where, and and are the upper and lower boundary, respectively. Bacteria will search for nutrition continuously as long as they need not to migrate. Migration in BCO algorithm is influenced by average energy degree, individual similarity, and chemotaxis efficiency. These factors altogether make up the migration condition. Pseudocode 2 shows the entire migration procedure.
3.5. Bacterial Colony Optimization
3.5.1. Implementation of Chemotaxis and Communication
As illustrated above, bacteria chemotaxis all life time can be divided into two models: tumbling and swimming. In the process of tumbling, a stochastic direction participates into actually swimming process. Therefore, turbulent director and optimal searching director altogether influence the search orientation in tumbling, update the positions of each bacterium as (3.4), whereas no turbulent director acceding in swimming process to affect the bacteria swimming toward optimal as (3.5) formulated: where and is the turbulent direction variance of the th bacterium. and are the globe best or personal best position of the th bacterium. is the chemotaxis step size. The BCO uses the adaptive chemotaxis step referred to Niu et al. [19, 20]: where is the maximal number of iterations, is the current number of iterations, and is the chemotaxis step of the th bacterium. With , the system becomes a special case of fixed chemotaxis step length, as the original proposed BFO algorithm. If , it is the linearly decreasing strategy of chemotaxis step. And otherwise, chemotaxis size is changing with a nonlinear decreasing strategy. As it is proved by Niu et al. in their chemotaxis step discussed papers [19, 20], simple low-dimension problems prefer to linear decreasing strategy of chemotaxis step, but, in high dimension multimodal complex problems, nonlinear decreasing strategies of chemotaxis step are more popular.
3.5.2. Implementation of Interactive Exchange
Interactive exchange in BCO can be divided into individual exchange and group exchange as described above. Individual exchange also can specify dynamic neighbor oriented (Figure 2(a)) and random oriented (Figure 2(b)). Group exchange means group oriented (Figure 2(c)). But in each generation, every bacterium has only one type of the exchange model to choose. As interactive exchange may affect the diversity of bacterial group, each bacterium has only one chance to exchange in each generation. Pseudocode 3 shows the process of interactive exchange.
3.5.3. Framework of Bacterial Colony Optimization
In BCO algorithm, artificial bacterial behaviors are executed based on given conditions responding to the dynamic environment. The procedure of chemotaxis, communication, reproduction, elimination, and migration is not premeditated, but determined only when certain given conditions are reached.
As shown in Figure 5, the process of reproduction, elimination, and migration is all independent without specific orders, where iteration times of BCO algorithm are the same as the frequency of chemotaxis and communication. It is consistent with the theoretical understanding of biological systems. The environment-based operation rules control the basic behavior of bacteria. Taking migration for example, Figure 5 also demonstrates that influence factors such as position, direction, and energy level can impact the migration process. So the rules of migration can be set by consulting the influence factors. As one of migration factors, position is related to group diversity, individual fitness, and so forth.
The overall procedure of bacterial colony optimization (BCO) is presented in Pseudocode 4.
4. Experiments and Results
4.1. Test Functions
To test the effectiveness of the new proposed BCO algorithm, twelve well-known test functions with 15 dimensions and 40 dimensions are adopted. Test problems include two unimodal functions , six multimodal functions , and four rotated multimodal functions . All the test functions used have a minimum function value of zero. The detailed description of the benchmark functions is listed in Appendix A.
4.2. Parameters Settings
To evaluate the performance of the proposed BCO, five other algorithms were used for comparisons: particle swarm optimization (PSO), genetic algorithm (GA), bacterial foraging optimization (BFO), bacterial foraging optimization with linear decreasing cemotaxis step (BFO-LDC), and bacterial foraging optimization with nonlinear decreasing chemotaxis step (BFO-NDC). The parameters used for these five algorithms were recommended from [4, 11, 19, 20, 23] or hand selected. The parameter setting of the previous algorithms on benchmark functions is summarized as Table 1. The population size of all algorithms used in our experiments was set at 100. The maximum number of iterations 2000 was applied to all the algorithms. For BCO, linearly decreasing chemotaxis step length is adopted. The lower step length is set at 0.01, and the upper step length is used. The test functions used in this paper are to find the minimum 0. All experiments were repeated for 20 runs.
4.3. Experimental Results and Discussions
4.3.1. Results for the 15-D Problems
Tables 2, 3, and 4 show the means and variances of the 20 runs of five algorithms on twelve test functions with 15 dimensions. And Figure 6 presents the convergence characteristics in terms of the best fitness values of median run of each algorithm for twelve unconstrained functions. In comparison with five introduced algorithms, the results in Tables 2~4 show that the proposed BCO performs significantly better than four other algorithms (GA, BFO, BFO-LDC, and BFO-NDC) in all test functions, and BCO generates better results than PSO in most of functions. Three exceptions in twelve are sphere, sum of power, and rotated Griewank function, and PSO can get the smaller values after iterations finished. From Table 2, PSO can get better performance in most of functions in this group. PSO is easy to implement in easy problems, BCO has to spend more time communicating and migrating, which result that the BCO could not converge as fast as PSO.
(a) Sphere function
(b) Rosenbrock function
(c) Sum of powers function
(d) Sin function
(e) Rastrigin function
(f) Griewank function
(g) Ackley function
(h) Weierstrass function
(i) RO-Rastrigin function
(j) RO-Griewank function
(k) RO-Ackley function
(l) RO-Weierstrass function
The means and variances of the median run of each algorithm on ~ multimodal functions with 15-D are presented in Pseudocode 3. From Pseudocode 3, it is observed that, for those four multimodal test problems, BCO achieves the best results compared with the other algorithms and it converges quickly (i.e., as shown in Figures 6(e)~6(h)). In this group, BCO obviously performs better than PSO. That is because PSO has the high chance of getting trapped in a local optimum in multimodal functions. In contrast, with the ability of migration, BCO has the best ability to surpass all approaches in keeping the tradeoff between the local exploitation and the global exploration. BCO could show better search ability in multimodal functions.
Pseudocode 4 illustrates the comparisons of the other four algorithms on the rotated benchmark functions (~). The proposed algorithm BCO is superior to any other algorithms (PSO, GA, BFO, BFO-LDC, and BFO-NDC) on the optimization of the optimization problems except the Rotated-Griewank function. From the values in Pseudocodes 3 and 4, BCO can converge to the best fitness value with the most times in the 15-D, which proves that BCO owns higher optimization capability for the multimodal complex problems.
Figure 6 shows the comparisons on the twelve unimodal, multimodal, and rotated functions in 15-D. This figure illustrates that the BCO could converge to the global optimum keeping a good diversity and high speed when it conducts the optimization of 15-D functions. From Figures 6(a)~6(c), Figures 6(e)~6(g) and others, BCO converges quicker than any other algorithms, which owes to the communication mechanism between bacterial individuals and bacterial groups. The fourth function Sin is a trigonometric function drawn from Figure 6(d), PSO gets the quickest convergence rate, but BCO finds the smallest optimum value after the maximum generations finished.
4.3.2. Results for the 40-D Problems
The experiments conducted on 15-D problems are repeated on the 40-D problems. Similarly to the case in 15-D, Tables 5~7 list the experimental results (i.e., the mean and standard deviations of the function values found in 20 runs) for each algorithm on twelve test functions. The average convergence results of benchmark functions on 40-D obtained in 20 runs are presented in Figure 16. From the results in Table 5, it is observed that the ranking of algorithms achieved is similar to the ranking in the 15-D problems. The new proposed approach BCO can find relatively optimum within 2000 generations and obtains the best results in the functions , and . Table 6 and Figures 7(e)~7(h) prove that the proposed algorithm can converge much faster to the best results than all other algorithms to the functions from to . As it concerns to the difficulty in multimodal functions ( to ), the proposed approach consistently finds the best minimum on the functions , , and in Table 7. Figures 7(i), 7(k), and 7(l) once again verify the fastest convergence rate of the proposed algorithm on those functions. Although it becomes more difficult in 40-D problems, BCO also achieves the best results compared with the other algorithms in most cases.