Substrates, enzymes, and products of the shikimate pathway.
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"6154",leadTitle:null,fullTitle:"Diabetes and Its Complications",title:"Diabetes and Its Complications",subtitle:null,reviewType:"peer-reviewed",abstract:"Diabetes is a complex, progressive disease, which is accompanied by several complications. 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This time variable allows one to derive other key system parameters such as channel occupancy time, new call blocking probability, and handoff call dropping probability. CHT depends on cellular shape, cell size, user’s mobility patterns, used handoff scheme, and traffic flow characteristics. Traffic flow characteristics are associated with unencumbered service time (UST), while the overall effects of cellular shape, users’ mobility, and handoff scheme are related to cell dwell time (CDT).
For convenience and analytical/computational tractability, the teletraffic analysis of mobile cellular networks has been commonly performed under the unrealistic assumption that CDT and/or CHT follow the negative exponential distribution (Lin et al., 1994; Hong & Rappaport, 1986). However, a plenty of evidences showed that these assumptions are not longer valid (Wang & Fan, 2007; Christensen et al., 2004, Fang, 2001, 2005; Orlik & Rappaport, 1998; Fang & Chlamtac, 1999; Fang et al., 1999; Alfa & Li, 2002; Rahman & Alfa, 2009; Soong & Barria, 2000; Yeo & Jun, 2002; Pattaramalai, et al., 2007). Recent papers have concluded that in order to capture the overall effects of users’ mobility, one needs suitable models for CDT distribution (Lin, 1994; Hong & Rappaport, 1986). In specific, the use of general distributions for modeling this time variable has been highlighted. In this research direction, some authors have used Erlang, gamma, uniform, deterministic, hyper-Erlang, sum of hyper-exponentials, log-normal, Pareto, and Weibull distributions to model the pdf of CDT; see (Wang & Fan, 2007; Fang, 2001, 2005; Orlik & Rappaport, 1998; Fang & Chlamtac, 1999; Fang et al., 1997, 1999; Rahman & Alfa, 2009; Pattaramalai et al., 2007, 2009; Hidata et al., 2002; Thajchayapong & Toguz, 2005; Khan & Zeghlache, 1997; Zeng et al. 2002; Kim & Choi, 2009) and the references therein. Fang in (Fang, 2001)) emphasizes the use of phase-type (PH) distributions for modeling CDT. The reason is twofold. First, PH distributions provide accurate description of the distributions of different time variables in wireless cellular networks, while retaining the underlying Markovian properties of the distribution. Markovian properties are essential in generating tractable queuing models for cellular networks. Second, there have been major advances in fitting PH distributions to real data. Among the PH probability distributions, the use of either Coxian or Hyper-Erlang distributions are of particular interest because their universality property (i.e, they can be used to approximate any non negative distribution arbitrarily close) (Soong & Barria, 2000; Fang, 2001).
Due to the discrepancy and the wide variety of proposed models, it appears mandatory to investigate the implications of the cell dwell time distribution on channel holding time characteristicsin mobile wireless networks. This is the topic of research of the present chapter. Let us describe the related work reported in this research direction.
In (Fang, 2001; Zeng et al. 2002), it is observed that, depending on the variance of CDT, the mean channel holding time for new calls (CHTn) can be greater than the mean channel holding time for handoff calls (CHTh). However, in these works, it is neither explained nor discussed the physical reasons for this observed behavior. This phenomenon (which is addressed in Section 3.1) and the lack of related published numerical results have motivated the present chapter.
Most of the previously published papers that have developed mathematical models for the performance analysis of mobile cellular systems considering general probability distribution for cell dwell time have either only presented numerical results for the Erlang (Wang & Fan, 2007; Fang et al., 1999; Rahman & Alfa, 2009; Kim & Choi, 2009) or Gamma distributions with shape parameter greater than one[1] - (Yeo & Jun, 2002; Fang, 2005), and/or only for the CHTh[1] - (Fang, 2001; Fang & Chlamtac, 1999), or have not presented numerical results at all (Fang, 2005; Alfa & Li, 2002; Soong & Barria, 2000). Thus, numerical results both for values of the coefficient of variation (CoV) of CDT greater than one and/or for the CHTn have been largely ignored. Exceptions of this are the papers (Orlik & Rappaport, 1998; Fang et al., b, 1997; Pattaramalai, et al., 2009).
On the other hand, probability distribution of CHT has been determined under the assumption of the staged distributions sum of hyper-exponentials, Erlang, and hyper-Erlang for the CDT (Orlik & Rappaport, 1998; Soong & Barria, 2000). However, to the best of the authors’ knowledge, probability distribution of CHT in mobile cellular networks with neither hyper-exponential nor Coxian distributed CDT has been previously reported in the literature.
In this Chapter, the statistical relationships among residual cell dwell time (CDTr), CDT, and CHT for new and handoff calls are revisited and discussed. In particular, under the assumption that UST is exponentially distributed and CDT is phase-type distributed, a novel algebraic set of general equations that examine the relationships both between CDT and CDTr and between CDT and channel holding times are obtained. Also, the condition upon which the mean CHTn is greater than the mean CHTh is derived. Additionally, novel mathematical expressions for determining the parameters of the resulting CHT distribution as functions of the parameters of the CDT distribution are derived for hyper-exponentially or Coxian distributed CDT.
A homogeneous multi-cellular system with omni-directional antennas located at the centre of each cell is assumed; that is, the underlying processes and parameters for all cells within the cellular network are the same, so that all cells are statistically identical.As mobile user moves through the coverage area of a cellular network, several variables can be defined: cell dwell time, residual cell dwell time, channel holding time, among others. These time variables are defined in the next section.
In this section the different time interval variables involved in the analytical model of a mobile cellular network are defined.
First, the unencumbered service time per call xs (also known as the requested call holding time (Alfa and Li, 2002) or call holding duration (Rahman & Alfa, 2009)) is the amount of time that the call would remain in progress if it experiences no forced termination. It has been widely accepted in the literature that the unencumbered service time can adequately be modeled by a negative exponentially distributed random variable (RV) (Lin et al., 1994; Hong & Rappaport, 1986). The RV used to represent this time is Xs and its mean value is
Now, cell dwell time or cellresidence timexd(j) is defined as the time interval that a mobile station (MS) spends in the j-th (for j = 0, 1, …) handed off cell irrespective of whether it is engaged in a call (or session) or not. The random variables (RVs) used to represent this time are Xd(j) (for j = 0, 1, …) and are assumed to be independent and identically generally phase-type distributed. For homogeneous cellular systems, this assumption has been widely accepted in the literature (Lin et al., 1994; Hong & Rappaport, 1986; Orlik & Rappaport, 1998; Fang & Chlamtac, 1999; Alfa & Li, 2002; Rahman & Alfa, 2009).
In this Chapter, cell dwell time is modeled as a general phase-type distributed RVwith the probability distribution function (pdf)
The residual cell dwell timexr is defined as the time interval between the instant that a new call is initiated and the instant that the user is handed off to another cell. Notice that residual cell dwell time is only defined for new calls. The RV used to represent this time is Xr. Thus, the probability density function (pdf) of Xr,
where E[Xd] and
Finally, we define channel holding time as the amount of time that a call holds a channel in a particular cell.In this Chapter we distinguish between channel holding times for handed off (CHTh) and channel holding time for new calls (CHTn). CHTh (CHTn) is represented by the random variable
The relationship between the probability distributions of CDT and CDTr is determined by the residual life theorem. In Table I some particular typically considered CDT distributions and the corresponding CDTr distributions obtained by applying the residual life theorem are shown.
Examples of corresponding distribution for Xr given the distribution of Xd.
The functional relationship between the moments of the residual cell dwell time and the cell residual time was obtained in (Kleinrock, 1975) applying the Laplace transform to the residual life theorem. That is,
This equation can be rewritten as
The n-th moment of the residual cell dwell time in terms of the moments of the cell dwell time can be obtained by deriving n times (equation 3) with negative argument and substituting s=0. Then (Kleinrock, 1975),
The mean residual cell dwell time as function of the moments of cell dwell time can be obtained as (Kleinrock, 1975)
In this way, the relationship between mean CDT and mean CDTr only depends on the value of the CoV of CDT. Thus, the mean CDTr is greater than the mean CDT (i.e., E{Xr} > E{Xd}) when the CoV of CDT is greater than one. This behavior (i.e., E{Xr} > E{Xd}) may seem to be counterintuitive due to the fact that, for a particular realization and by definition, CDTr cannot be greater than CDT[1] -. This occurs because in such conditions there is a high variability on the cell dwell times in different cells and it is more probable to start new calls on cells where users spent more time. Then, residual cell dwell times tend to be greater than the mean CDT. This phenomenon that may seem to be counterintuitive is now explained and mathematically formulated in this Chapter.
Channel holding times for handed off and new calls (denoted by XC(h) and XC(N), respectively) are given by the minimum between UST and CDT or CDTr, respectively. The CDF of the CHTh and CHTn are, respectively, given by
Due to the fact that the Laplace transform of the pdf of both UST and CDTr are rational functions, the Laplace transform of the pdf of CHTn can be obtained using the Residue Theorem as follows (Wang & Fan, 2007)
where
Under the condition that UST is general phase type (PH) distributed, the authors of (Alfa & Li, 2002) prove that the CDT is PH distributed if and only if the CHTn is PH distributed or the CHTh is PH distributed.
The probability distributions of CHTn and CHTh for different staged probability distributions of CDT assuming that the UST is exponentially distributed are shown in Table II. The first entry of this table is a well known result[1] -. In (Soong & Barria, 2000), it was shown that when CDT has Erlang or hyper-Erlang distribution, channel holding times have the uniform Coxian and hyper-uniform Coxian distribution, respectively.Uniform Coxian is a special case of the Coxian distribution where all the phases have the same parameter (Perros & Khoshgoftaar, 1989). The hyper-uniform Coxian distribution is a mixture of uniform Coxian distributions.
Examples of corresponding distributions for Xc(N) and Xc(h)
Next, it is shown that when the UST is exponentially distributed and CDT has hyper-exponential distribution of order n, the distribution of CHTh has also a hyper-exponential distribution of order n. Similarly, when CDT has Coxian distribution of order n, the distribution of CHTn has also a Coxian distribution of order n.
Considering that CDT has a hyper-exponential pdf of order n given by
For exponentially distributed UST and using (4), the CDF of the CHTh can be expressed as follows
This expression corresponds to a hyper-exponential distribution of order n with phase parameters μ + ηi and probabilities Pi of choosing stage i (for i = 1, …, n).
As the CDTr is hyper-exponentially distributed when CDT has hyper-exponential distribution, the CHTn is also hyper-exponentially distributed. In this case, the probability of choosing stage i (for i = 1, …, n) is given by
Considering that cell dwell time has an m-th order Coxian distribution (which diagram of phases is shown in Fig. 1) with Laplace transform of its pdf given by
where
(1- αi) represents the probability of passing from the i-th phase to the (i+1)-th one.
Diagram of phases of the considered Coxian distribution of order m for modeling cell dwell time.
For exponentially distributed UST and using (9), the Laplace transforms of the pdf of CHTh and CHTn are given by
Replacing (13) into (15), it can be written as
where
for i = 1, …, m. Then, CHTh has also a Coxian distribution of order m but with parameters (μ + ηi), for i = 1, …, m.
On the other hand, the Laplace transform of the residual cell dwell time can be shown to be given by
where
for j = 1, …, m(m+1)/2. Substituting (19) into (16), Laplace transform of CHTn can be written as
where
Equation (23) corresponds to the Laplace transform of a generalized Coxian pdf.
The above analytical results show that CHTh (CHTn) has the same probability distribution as CDT (CDTr) but with different parameters of the phases, probabilities of reaching the absorbing state after each phase, and probabilities of choosing each stage. The detailed derivation of the last entry of Tables I and II (i.e., when cell dwell time has generalized Coxian distribution) is addressed in (Corral-Ruiz et al., a, 2010).
Using (15) and (16) it is straightforward to show that the mean values of CHTn and CHTh are, respectively, given by
At this point, it is important to mention that authors in (Fang, 2001; Zeng et al., 2002) stated that, depending on the variance of CDT, the mean CHTn can be greater than the mean CHTh. However, it was neither explained nor discussed the physical reasons for this observed behavior. This behavior occurs because the residual cell dwell times tend to increase as the variance of cell dwell time increases, as it was explained above.
Using (25) and (26), the condition for which the mean CHTn is greater that the mean CHTh, that is,
can be easily found. This condition is given by
Thus, the relationship between the mean new and handoff call channel holding times is determined by the mean values of both CDT and UST and by the Laplace transform of the pdf of CDT evaluated at the inverse of the mean UST.
Finally, in a similar way, the squared coefficient of variation for CHTn and CHTh can be shown to be given, respectively, by
It can be shown that the n-th moments for new and handoff call channel holding times are given, respectively, by
In this section, numerical results on how the distribution of cell dwell time (CDT) affects the characteristics of channel holding time (CHT) are presented. We use different distributions to model CDT, say, negative-exponential, constant (deterministic), Pareto with shape parameter α in the range (1, 2] (i.e., when infinite variance is considered), Pareto with α>2 (i.e., when finite variance is considered), log-normal, gamma, hyper-Erlang of order (2,2), hyper-exponential of order 2, and Coxian of order 2. Three different mobility scenarios for the numerical evaluation are assumed: E{Xd}=5E{Xs} (low mobility), E{Xd}=E{Xs} (moderate mobility), and E{Xd}=0.2E{Xs} (high mobility). In the plots of this section we use E{Xs}=180 s. In our numerical results, the effect of CoV and skewness of CDT on CHT characteristics is investigated. In the plots presented in this section, “HC” and “NC” stand for channel holding time for handoff calls (CHTh) and channel holding time for new calls (CHTn), respectively.
Fig. 2 plots the mean value of both CHTn and CHTh versus the mean value of CDT when it is modeled by negative-exponential (EX), constant, and Pareto with 1<2 distributions. It is important to remark that all of these distributions are completely characterized by their respective mean values. As expected, Fig. 2 shows that, for the case when CDT is exponentially distributed, mean CHTn is equal to mean CHTh. An interesting observation on the results shown in Fig. 2 is that, irrespective of the mean value of CDT, there exists a significant difference between the mean value of CHTn when CDT is modeled as exponential distributed RV and the corresponding case when it is modeled by a heavy-tailed Pareto distributed RV (this behavior is especially true for the case when α=1.1). Notice, however, that this difference is negligible for the case when α=2 and high mobility scenarios (say, E{Xd}<50 s) are considered. Similar behaviors are observed if mean CHTh is considered. Consequently, for high mobility scenarios where CDT can be statistical characterized by a Pareto distribution with shape parameter close to 2, the exponential distribution represents a suitable model for the CDT distribution. Fig. 2 also shows that, for
Mean new and handoff call channel holding time for deterministic, negative exponentially, and Pareto distributed CDT against the mean CDT.
a given value of the mean CDT and considering the case when CDT is Pareto distributed with α=1.1 (α=2), mean CHTn always is greater (lower) than mean CHTh. This behavior can be explained by the combined effect of the following two facts. First, as α comes closer to 1 (2), the probability that CDT takes higher values increases (decreases). This fact contributes to increase (reduce) the mean CHTh. Second, in general, new calls are more probable to start on cells where users spent more time and, as α comes closer to 1, this probability increases. This fact contributes to increase mean CHTn relative to the mean CHTh. Then, the combined effect is dominated by the first (second) fact as α comes closer to 2 (1). This leads us to the behavior explained above and illustrated in Fig.2. It may be interesting to derive the condition upon which the mean CHTn is greater than the mean CHTh when CDT is heavy-tailed Pareto distributed. This represents a topic of our current research.
Fig. 3 plots the mean value of both CHTn and CHTh versus the CoV of CDT when it is modeled by Pareto with shape parameter α>2, lognormal, and Gamma distributions; all of them with mean value equal to 180 s. It is important to remark that all of these distributions are completely characterized by their respective first two moments. Fig. 3 shows that both mean CHTn and mean CHTh are highly sensitive to the type of distribution of CDT; this fact is especially true for CoV>2. Notice that, for the particular case when CoV=0, the mean values of both CHTn and CHTh are identical to the corresponding values for the case when CDT is deterministic with mean value equals 180 s, as expected. Fig. 3 also shows that, for values of CoV of CDT greater than 1 (1.2), mean CHTn is greater that mean CHTh when CDT is Gamma (log-normal) distributed. On the other hand, when CDT is Pareto distributed and irrespective of the value of its CoV, CHTh always is greater that mean CHTn. This behavior is mainly due to the heavy-tailed characteristics of the Pareto distribution.
Mean new and handoff call channel holding time for gamma, log-normal, and Pareto distributed cell dwell time versus CoV of cell dwell time.
Figs. 4, 5, and 6 (7, 8, and 9) plot the mean value (CoV) of both CHTn and CHTh versus both the CoV and skewness of CDT when it is modeled by hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions, respectively. It is important to remark that all of these distributions are completely characterized by their respective first three moments. Results of (Johnson & Taaffe, 1989; Telek & Heindl, 2003) are used to calculate the parameters of these distributions as function of their first three moments. In Figs. 4 to 9, two different values for the mean CDT are considered: 36 s (high mobility scenario) and 900 s (low mobility scenario). From Figs. 2, 5 and 6 the following interesting observation can be extracted. Notice that, for the case when CDT is modeled by either hyper-exponential or Coxian distributions and irrespective of the mean value of CDT, the particular scenario where skewness and CoV of CDT are, respectively, equal to 2 and 1, corresponds to the case when CDT is exponential distributed (in the exponential case mean CHTn and mean CHTh are identical).
Mean CHTn and mean CHTh for hyper-Erlang distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.
Mean CHTn and mean CHTh for hyper-exponentially distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.
Mean CHTn and mean CHTh for Coxian distributed cell dwell time versus CoV and skewness of cell dwell time, with the mean CDT as parameter.
CoV of CHTn and CHTh for hyper-Erlang distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.
CoV of CHTn and CHTh for hyper-exponential distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.
CoV of CHTn and CHTh for Coxian distributed CDT versus CoV and skewness of cell dwell time, with the mean cell dwell time as parameter.
On the other hand, Fig. 4 shows that the case when hyper-Erlang distribution with skewness equals 2 and CoV equals 1 is used to model CDT does not strictly correspond to the exponential distribution; however, the exponential model represents a suitable approximation for CDT in this particular case. From Figs. 4 to 9, it is observed that the qualitative behavior of mean and CoV of both CHTn and CHTh is very similar for all the phase-type distributions under study. The small quantitative difference among them is due to moments higher than the third one. Analyzing the impact of moments of CDT higher than the third one on channel holding time characteristics represents a topic of our current research.
From Fig. 10 is observed that the difference among the mean values of CHTn and CHTh is strongly sensitive to the CoV of the CDT, while it is practically insensitive to the skewness of the CDT. This difference is higher for the case when the CDT is modeled as hyper-exponential distributed RV compared with the case when it is modeled as hyper-Erlang distributed RV. Also, it is observed that this difference remains almost constant for the entire range of values of the CoV of the CDT.
Difference among the mean values of new and handoff call channel holding times for hyper-Erlang and hyper-exponential distributed cell dwell time versus CoV and skewness of cell dwell time, for the moderate-mobility scenario.
Finally, in Fig. 11 the mean channel holding time for new and handoff calls considering the gamma, hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions for the cell dwell time are shown for different values of the coefficient of variation. The numerical results shown in Fig. 11 are obtained by equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 11, it is observed that for the hyper-exponentialand Coxian distributions practically the same results are obtained for the mean channel holding time for both new and handoff calls. The differences among the other distributions are due to the fact that they differ on the higher order moments. To show this, the forth standardized moment (i.e., excess kurtosis) of the different distributions is shown in Fig. 12 for different values of the coefficient of variation, equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 12, it is observed that the hyper-exponentialand Coxian distributions practically have the same value of excess kurtosis but this differs for that of the gamma and hyper-Erlang distributions. The gamma distribution shows the more different value of the excess kurtosis and, therefore, for this distribution the more different values of the mean channel holding times in Fig. 11 are obtained. Then, it could be necessary tocapture more than three moments, even though the lower ordermoments dominate in importance. Similar conclusion was drawn in (Gross &Juttijudata, 1997).
Mean new and handoff call channel holding time for gamma, hyper-exponential (2), hyper-Erlang (2,2) and Coxian (2) distributed cell dwell time versus CoV of cell dwell time.
Kurtosis of cell dwell time for gamma, hyper-exponential (2), Coxian (2) and hyper-Erlang (2,2) distributed cell dwell time versus CoV of cell dwell time.
In this Chapter, under the assumption that unencumbered service time is exponentially distributed, a set of novel general-algebraic equations that examines the relationships between cell dwell time and residual cell dwell time as well as between cell dwell time and new and handoff channel holding times was derived. This work includes relevant new analytical results and insights into the dependence of channel holding time characteristics on the cell dwell time probability distribution. For instance, we found that when cell dwell time is Coxian or hyper-exponentially distributed, channel holding times are also Coxian or hyper-exponentially distributed, respectively. Also, our analytical results showed that the mean and coefficient of variation of the new and handoff call channel times depend on Laplace transform and first derivative of the Laplace transform of the probability density function of cell dwell time evaluated at the inverse of the mean unencumbered service time as well as on the mean of both cell dwell time and unencumbered service time. Additionally, we derive the condition upon which the mean new call channel holding time is greater than the mean handoff call channel holding time. Similarly, the condition upon which the mean residual cell dwell time is greater than the mean cell dwell time was also derived. To the best authors’ knowledge, this phenomenon that may seem to be counterintuitive has been explained and mathematically formulated in this Chapter. We believe that the study presented here is important for planning, designing, dimensioning, and optimizing of mobile cellular networks.
The secondary metabolism is a biosynthetic source of several interesting compounds useful to chemical, food, agronomic, cosmetics, and pharmaceutical industries. The secondary pathways are not necessary for the survival of individual cells but benefit the plant as a whole [1]. Another general characteristic of secondary metabolism is that found in a specific organism, or groups of organisms, and is an expression of the individuality of species [2]. The secondary metabolism provides chemical diversity to organic molecules with low molecular weight that are related by the respective pathways; such organic molecules are called secondary metabolites. The secondary metabolites are often less than 1% of the total carbon in plant molecules [3]. These organic molecules isolated from terrestrial plants are the most studied, and their syntheses have an important role in the protection against pathogens, unfavorable temperature and pH, saline stress, heavy metal stress, and UVB and UVA radiation [3]. Secondary metabolism reflects plant environments more closely than primary metabolism [4]. There are three principal kinds of secondary metabolites biosynthesized by plants: phenolic compounds, terpenoids/isoprenoids, and alkaloids and glucosinolates (nitrogen- or sulfur-containing molecules, respectively) [5]. Phenolic compounds are biosynthesized by the shikimate pathway and are abundant in plants. The shikimate pathway, in plants, is localized in the chloroplast. These aromatic molecules have important roles, as pigments, antioxidants, signaling agents, electron transport, communication, the structural element lignan, and as a defense mechanism [6], Figure 1. The seven steps of the shikimate pathway and the metabolites for branch point are described in this chapter, as factors that induce the synthesis of phenolic compounds in plants. Some representative examples that show the effect of biotic and abiotic stress on the production of phenolic compounds in plants are discussed.
\nPhenolic compound biosynthesis promoted by biotic and abiotic stresses (e.g., herbivores, pathogens, unfavorable temperature and pH, saline stress, CO2, O3, heavy metal stress, and UVB and UVA radiation).
The shikimate biosynthesis pathway provides precursors for aromatic molecules in bacteria, fungi, apicomplexan, and plants, but not in animals [2, 7]. Shikimic acid is named after the highly toxic Japanese shikimi (Illicium anisatum) flower from which it was first isolated [8]. This biochemical pathway is a major link between primary and secondary metabolism in higher plants [6]. In microorganisms, the shikimate pathway produces aromatic amino acids L-phenylalanine (L-Phe), L-tyrosine (L-Tyr), and L-tryptophan (L-Trp), molecular building blocks for protein biosynthesis [9]. But in plants, these aromatic amino acids are not only crucial components of protein biosynthesis; they also serve as precursors for diverse secondary metabolites that are important for plant growth [10]. These secondary metabolites are called phenolic compounds and are synthesized when needed by the plant [11]. These molecules play an important role in the adaptation of plants to their ecosystem, and their study advances biochemical techniques and molecular biology [3, Bourgaud]. The principal aromatic phenolic compounds synthesized from L-Phe and L-Tyr are cinnamic acids and esters, coumarins, phenylpropenes, chromones (C6-C3), stilbenes, anthraquinones (C6-C2-C6), chalcones, flavonoids, isoflavonoids, neoflavonoids (C6-C3-C6), and their dimers and trimers, respectively (C6-C3-C6)2,3, lignans, neolignans (C6-C3)2, lignans (C6-C3)n, aromatic polyketides, or diphenylheptanoids (C6-C7-C6) [12]. L-Trp is a precursor of alkaloids in the secondary metabolism [2]. Additionally, diverse hydroxybenzoic acids and aromatic aldehydes (C6-C1) are biosynthesized via branch points in the shikimate pathway, Figure 2. Phenolic compounds biosynthesized from the shikimate pathway have structural versatility.
\nThe shikimic and chorismic acids are the common precursors for the synthesis of L-Phe, L-Tyr, and L-Trp and diverse phenolic compounds.
The shikimate pathway consists of seven sequential enzymatic steps and begins with an aldol-type condensation of two phosphorylated active compounds, the phosphoenolpyruvic acid (PEP), from the glycolytic pathway, and the carbohydrate D-erythrose-4-phosphate, from the pentose phosphate cycle, to give 3-deoxy-D-arabino-heptulosonic acid 7-phosphate (DAHP), Figure 3. The seven enzymes that catalyze the pathway are known: 3-deoxy-D-arabino-heptulosonate-7-phosphate synthase (DAHPS; EC 4.1.2.15, now EC 2.5.1.54), 3-dehydroquinate synthase (DHQS; EC 4.2.3.4), 3-dehydroquinate dehydratase/shikimate dehydrogenase (DHQ/SDH; EC 4.2.1.10/EC 1.1.1.25), shikimate kinase (SK; EC 2.7.1.71), 5-enolpyruvylshikimate 3-phosphate synthase (EPSPS; EC 2.5.1.19), and chorismate synthase (CS; EC 4.2.3.5) [13], Table 1.
\nShikimate pathway.
Reaction step | \nSubstrate | \nEnzyme/cofactor | \nProduct | \n
---|---|---|---|
1 | \nPhosphoenolpyruvate (PEP), erythrose-4-phosphate | \n3-Deoxy-D-arabino-heptulosonate-7-phosphate synthase (DAHPS; EC 4.1.2.15, now EC 2.5.1.54)/Co2+, Mg2+ or Mn2+ [15] | \n3-Deoxy-D-arabino-heptulosonic acid 7-phosphate (DAHP), Pi | \n
2 | \n3-Deoxy-D-arabino-heptulosonic acid 7-phosphate (DAHP) | \n3-Dehydroquinate synthase DHQS (EC. 4.2.3.4)/Co2+, NAD+ [15, 16] | \n3-Dehydroquinic acid (DHQ), Pi | \n
3 | \n3-Dehydroquinic acid (DHQ) | \n3-Dehydroquinate dehydratase (DHQ dehydratase EC 4.2.1.10) [15] | \n3-Dehydroshikimic acid (DHS), H2O | \n
4 | \n3-Dehydroshikimic acid (DHS), NADPH + H+ | \nShikimate dehydrogenase (SDH; EC 1.1.1.25) [18, 19, 20, 21] | \nShikimic acid, NADP+ | \n
5 | \nShikimic acid, ATP | \nShikimate kinase enzyme (SK; EC 2.7.1.71) | \nShikimic acid 3-phosphate (S3P), ADP | \n
6 | \nShikimic acid 3-phosphate (S3P), PEP | \n5-Enolpyruvylshikimate 3-phosphate synthase, also called aroA enzyme (EPSPS; EC 2.5.1.19) [25] | \n5-Enolpyruvylshikimate 3-phosphate (EPSP), Pi | \n
7 | \n5-Enolpyruvylshikimate 3-phosphate (EPSP) | \nChorismate synthase (CS; EC 4.2.3.5)/FMNH2 [2, 19, 30, 31] | \nChorismic acid, Pi | \n
Substrates, enzymes, and products of the shikimate pathway.
Pi, phosphate; NAD+, oxidized nicotinamide adenine dinucleotide; NADPH, reduced nicotinamide adenine dinucleotide phosphate; FMNH2, reduced flavin mononucleotide.
The shikimate pathway has special characteristics that are present only in bacteria, fungi, and plants. The absence of the pathway in all other organisms provides the enzymes catalyzing these reactions with potentially useful targets for the development of antibacterial agents and herbicides. For example, 5-enolpyruvylshikimate 3-phosphate synthase (EPSP-synthase) catalyzes the transfer of the enolpyruvyl (carboxyvinyl) moiety from PEP to shikimic acid 3-phosphate (S3P) [6].
\nIn the second reaction step, DAHP loses phosphate (Pi); the enolic-type product is cyclized through a second aldol-type reaction to produce 3-dehydroquinic acid (DHQ). The 3-dehydroquinate synthase (DHQS) catalyzes this cyclization in the shikimate pathway. The DHQ dehydrates to produce 3-dehydroshikimic acid (DHS) (3-dehydroquinate dehydratase); this compound has a conjugated double carbon-carbon, Figure 3. The protocatechuic and the gallic acids (C6-C1) are produced by branch-point reactions from DHS [2]. The fourth step in the pathway is a reduction reaction of DHS with reduced nicotinamide adenine dinucleotide phosphate (NADPH), Figure 3. The fifth section of the pathway is the activation of shikimic acid with adenosine triphosphate (ATP) (shikimate kinase, SK) to make shikimic acid 3-phosphate (S3P). The sixth chemical reaction is the addition of PEP to S3P to generate 5-enolpyruvylshikimic acid 3-phosphate; the enzyme that catalyzes this reaction step, 5-enolpyruvylshikimate 3-phosphate synthase (EPSPS), has been extensively studied. The reason for this interest is because glyphosate [N-(phosphonomethyl)glycine] is a powerful inhibitor of EPSPS [2], so glyphosate has been used as a broad-spectrum systemic herbicide. It is an organophosphorus molecule, phosphonic acid, and glycine derivative that has a similar molecular structure to PEP, Figure 4.
\nPEP and glyphosate (powerful inhibitor of the 5-enolpyruvylshikimate 3-phosphate synthase, EPSPS).
The last reaction step of the shikimate pathway is the production of chorismic acid from catalytic action on the chorismate synthase (CS). This reaction is a 1,4-trans elimination of Pi, to yield the conjugated molecule, chorismic acid, Figure 3.
\nThe first reaction of the shikimate pathway is an aldol-type condensation of PEP and carbohydrate erythrose-4-P, to give 3-deoxy-D-arabino-heptulosonic acid 7-phosphate (DAHP), Figures 3 and 5. A new stereogenic center is generated in the condensation product DAHP catalyzed by the 3-deoxy-D-arabino-heptulosonate-7-phosphate synthase enzyme (DAHPS; EC 4.1.2.15, now EC 2.5.1.54). Results of enzymatic kinetic and labeled PEP with tritium (Z)-[3-3H] PEP suggest that the nucleophilic attack of PEP is from the Si face of PEP to the Re face of the carbonyl group of D-erythrose-4-P, Figure 5 [14]. Two isoenzymes of DAHPS have been found for the catalysis of this first reaction step. One isozyme needs only Mn2+, and the other, either Co2+, Mg2+, or Mn2+ for the catalysis [15].
\nStereochemistry of the condensation reaction of (Z)-[3-3H]PEP and D-erythrose-4-phosphate by DAHP synthase [14].
The second reaction of the shikimate pathway is an intramolecular aldol-type reaction cyclization, where the enol (C6-C7) of DAHP nucleophilically attacks the carbonyl group (C2), to produce a six-member cycle, the 3-dehydroquinic acid (DHQ), Figures 3 and 6. The enzyme that catalyzes this reaction, 3-dehydroquinate synthase DHQS (EC. 4.2.3.4), is a carbon-oxygen lyase enzyme that requires Co2+ and bound oxidized nicotinamide adenine dinucleotide (NAD+) as cofactors [15, 16]. The Co2+ is essential for the catalytic activity of DHQS. Bender et al. [16] found that DHQS, from Escherichia coli, is a monomeric metalloenzyme that contains tightly bound Co2+, and DHQS is deactivated with ethylenediaminetetraacetic acid (EDTA). The presence of the substrate (DAHP) blocks the inactivation by EDTA. The NAD+ cofactor dissociates form the DHQS enzyme rapidly in the presence of DAHP [16]. The reaction mechanism of the enzyme-catalyzed conversion of DAHP to DHQ involves five transformations from the DAHP hemiketal form, a pyranose: (1) oxidation of the hydroxyl at C5 adjacent to the lost proton that requires NAD+ (NAD+ need never dissociate from the active site), (2) the elimination of Pi of C7 to make the α,β-unsaturated ketone, (3) the reduction of C5 with NADH + H+, (4) the ring opening of the enol to yield an enolate, and (5) the intramolecular aldol-like reaction to produce DHQ. All five-reaction steps occur through the function of DHQS, Figure 6.
\nReaction mechanism of DAHP (hemiketal form) to 3-dehydroquinic acid (DHQ) by 3-dehydroquinate synthase DHQS (EC. 4.2.3.4) [16].
The reduction reaction of DHQ leads to quinic acid at this branch point in the shikimate pathway. Quinic acid is a secondary metabolite that is free, forming esters or as part of alkaloids such as quinine. Quinic acid is found in high quantities in mature kiwi fruit (Actinidia chinensis and other species of Actinidia) and is a distinguishing characteristic of fresh kiwi fruit [7]. Also, the quinic acid is abundant in roasted coffee [17].
\nThe third and fourth reaction steps of the shikimate pathway are catalyzed by a bifunctional enzyme: 3-dehydroquinate dehydratase/shikimate dehydrogenase (DHQ dehydratase/SDH; EC 4.2.1.10/EC 1.1.1.25). The DHQ dehydratase enzyme is a hydro-lyase kind, and the SDH is an oxidoreductase enzyme. The DHQ dehydratase, in the third reaction step, converts DHQ into 3-dehydroshikimic acid (DHS) by eliminating water, and this reaction is reversible, Figure 7. The DHS is converted to shikimic acid in the fourth reaction step, by the reduction of the carbonyl group at C-5 by the catalytic action of SDH with NADPH, Figure 3. The biosynthesis of DHS is a branch point to shikimic acid and to the catabolic quinate pathway. If the DHS dehydrates, it produces protocatechuic acid (C6-C1) or gallic acid, Figure 3. Gallic acid (C6-C1) is a hydroxybenzoic acid that is a component of tannins [2].
\nReaction mechanism to produce 3-dehydroshikimic acid (DHS) by type I DHQ dehydratase enzyme [21].
Two structurally different kinds of 3-dehydroquinate dehydratase are known: type I (not heat-stable) and type II (heat-stable). Type I enzyme is present in bacteria and higher plants, and type II is found in fungi, which have both types of enzymes [18, 19]. The catalytic mechanism of the type I DHQ dehydratase has been detected by electrospray MS [20]. This catalytic mechanism involves the amino acid residue Lys-241 that forms a Schiff base with the substrate and product, Figure 7 [21]. The fourth step is the reduction of DHS with NADPH that enantioselectively reduces the carbonyl of the ketone group of DHS to produce shikimic acid (shikimate dehydrogenase, SDH), Figure 3.
\nSigh and Christendat [22] reported the crystal structure of DHQ dehydratase/SDH from the plant genus Arabidopsis. The crystal structure has the shikimate bound at the SDH and the tartrate molecule at the DHQ dehydratase. The studies show that Asp 423 and Lys 385 are key catalytic amino acids and Ser 336 is a key-binding group.
\nThe shikimate kinase enzyme (SK; EC 2.7.1.71) catalyzes the phosphorylation of the shikimic acid, the fifth chemical reaction of the shikimate pathway, and the products are shikimic acid 3-phosphate (S3P) and ADP, Figures 3 and 8. Shikimic acid is phosphorylated with ATP in the 5-hydroxyl group of shikimic acid. SK is an essential enzyme in several bacterial pathogens and is not present in the human cell; therefore the SK enzyme has been classified as a protein target for drug design, especially for chemotherapeutic development of antitubercular drugs [23, 24].
\nPhosphorylation of shikimic acid with ATP.
The 5-enolpyruvylshikimate 3-phosphate synthase, also called aroA enzyme (EPSPS; EC 2.5.1.19), catalyzes the condensation of PEP to the 5-hydroxyl group of S3P in the sixth reaction of the shikimate pathway to form 5-enolpyruvylshikimate 3-phosphate (EPSP). The reaction mechanism involves the protonation of PEP to subsequent nucleophilic attack of the hydroxyl at C-5 of S3P to form an intermediate that loses Pi to form EPSP, Figure 9 [25].
\nReaction mechanism of the condensation of S3P with PEP by EPSPS (EC 2.5.1.19) to form EPSP [25].
EPSPS is the most studied enzyme of the shikimate pathway because it plays a crucial role in the penultimate step. If this enzyme is inhibited, there is an accumulation of shikimic acid [26], and the synthesis of aromatic amino acid is disabled, leading to the death of the plant [27]. Therefore, EPSPS is used as a target for pesticides, like glyphosate, Figure 4, the active ingredient in the herbicides RoundUp™, Monsanto Chemical Co., and Touchdown™, Syngenta. Glyphosate (N-(phosphonomethyl)glycine) inhibits EPSPS and is a potent nonselective herbicide that mimics the carbocation of PEP and binds EPEPS competitively [28]. Because the glyphosate is nonselective and kills food crops, there is interest in finding glyphosate-tolerant genes for genetically modified crops [29]. Two types of EPSPS enzymes have been identified: type I EPSPS (sensitive to glyphosate) identified mostly in plants and bacteria and type II EPSPS (nonsensitive to glyphosate and has a high affinity for PEP), found in some bacteria [27].
\nThe seventh and last reaction step of the shikimate pathway is the 1,4-trans elimination of the Pi group at C-3 from EPSPS to synthetize chorismic acid. This last step is catalyzed by chorismate synthase (CS; EC 4.2.3.5) that needs reduced flavin mononucleotide (FMNH2) as a cofactor that is not consumed [2, 19]. The FMNH2 transfers an electron to the substrate reversibly [30]. Spectroscopic techniques and kinetic isotope effect studies suggest that a radical intermediate in a non-concerted mechanism is developed [30, 31], Figure 10. Chorismic acid, the final molecule of the shikimate pathway, is a key branch point to post-chorismic acid pathways, to obtain L-Phe, L-Tyr, and L-Trp, Figure 2. L-Phe is the substrate to phenylpropanoid and flavonoid pathways [13].
\nReaction of mechanism to yield chorismic acid by chorismate synthase [30].
The expression of phenolic compounds is promoted by biotic and abiotic stresses (e.g., herbivores, pathogens, unfavorable temperature and pH, saline stress, heavy metal stress, and UVB and UVA radiation). UV radiation is divided into UVC (≤280 nm), UVB (280–320 nm), and UVA (300–400 nm). UVA and UVB radiation are transmitted through the atmosphere; all UVC and some UVB radiation (highly energetic) are absorbed by the Earth’s ozone layer. This accumulation is explained by the increase in enzymatic activity of the phenylalanine ammonia-lyase and chalcone synthase enzymes, among others [12]. Studies have been done about the increase of phenolic compounds, such as anthocyanins, in plants when they are exposed to UVB radiation [13]. Another study demonstrates that UVB exposure enhances anthocyanin biosynthesis in “Cripps pink” apples (Malus x domestica Borkh.) but not in “Forelle” pears (Pyrus communis L.) [32]. This effect may be due to UV radiation exposure and the cultivar of the plants studied. It is known that if plants are under stress, they accumulate phenolic compounds.
\nThe increase in phenolic compounds in blueberry (Vaccinium corymbosum) plantlets cultivated in vitro exposed to aluminum (Al) and cadmium (Cd) has also been studied. These heavy metals cause high toxicity in plants, because they increase the oxidative stress by the production of reactive oxygen species (ROS). The authors of the study suggest that the phenolic compounds, specifically chlorogenic and ellagic acids, Figure 11, reduce the ROS in blueberry plants [33].
\nChemical structure of chlorogenic (C6-C3) and ellagic (C6-C1) acids.
An interesting study was carried out in 2011 by Mody et al., where they studied the effect of the resistance response of apple tree seedlings (Malus x domestica) to a leaf-chewing insect (Spodoptera littoralis) [34]. The authors found a significant herbivore preference for undamaged plants (induced resistance) was first observed 3 days after herbivore damage in the most apical leaf. Also, the results showed higher concentrations of the flavonoid phlorizin, Figure 12, in damaged plants than undamaged plants. This indicates that insect preference for undamaged apple plants may be linked to phlorizin, which is the main secondary metabolite of the phenolic type in apple leaves.
\nChemical structure of phlorizin (C6-C3).
Knowledge of the biosynthetic pathway of shikimic acid leads to understanding the reaction mechanisms of enzymes and thus discovering antimicrobials, pesticides, and antifungals. Studies with isotopic labeling of substrates, the use of X-ray diffraction, nuclear magnetic resonance (NMR), mass spectrometry (ES), biotechnology, as well as organic synthesis have contributed to explaining the shikimate pathway. Although the seven steps of the biosynthetic pathway are elucidated, these metabolites are the precursors of phenolic compounds, more complex molecules that are necessary for the adaptation of plants to the environment. So, the shikimate pathway is the basis for the subsequent biosynthesis of phenolic compounds. There is scientific interest in continuing to investigate the biosynthesis of phenolic compounds from several points of view: pharmaceuticals, agronomy, chemical and food industries, genetics, and health.
\nThe authors thank Carol Ann Hayenga for her English assistance in the preparation of this manuscript. The Technological University of the Mixteca provided support.
\nThe authors have no conflict of interest to declare and are responsible for the content and writing of the manuscript.
This chapter does not contain any studies with human participants or animals performed by any of the authors.
\n.
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