12,290 research outputs found
Multifunctional mesoporous silica nanospheres for biosensor, stimuli-responsive controlled-release drug delivery carriers and gene transfection vectors
Structurally well-defined mesoporous silica materials with tunable pore size and narrow pore-size distribution have attracted much attention for their potential applications in sensing, drug delivery and gene transfer. A series of mesoporous silica nanosphere-based (MSN) nano-devices were synthesized and utilized for the aforementioned applications.;For sensing application, we provide a first example in utilization of a multi-functionalized mesoporous silica material to selectively detect dopamine. A second application, rooted in this work, is the selective detection of amino-containing neurotransmitters under physiological condition.;The drug delivery applications have been built around a controlled release drug delivery device. In the original conception, the system is using surface-derivatized cadmium sulfide (CdS) nanocrystals as chemically removable caps to encapsulate several drug molecules and neurotransmitters inside the organically functionalized MSN mesoporous framework. To study the cap-dependence release profile, adenosine 5-triphosphate (ATP) molecules were encapsulated in the MSNs followed by capping of the mesopores with chemically removable caps, such as CdS nanoparticles and poly(amido amine) dendrimers (PAMAM). A luciferase chemiluminescence imaging method was employed to investigate the kinetics and mechanism of the ATP release with various disulfide-reducing agents as uncapping triggers in real time. By varying the chemical nature of the cap and trigger molecules in our MSN system, we observed that the release profiles could be regulated.;Further, the PAMAM dendrimers of low generations were utilized as MSN caps for drug delivery. The biocompatibility and delivery efficiency of the later MSN system in interaction with HeLa cervical cancer cells were demonstrated. The system renders the possibility to serve as a universal transmembrane carrier for intracellular drug delivery and imaging applications. This is the first uptake study of MCM-41 type mesoporous silicas into eukaryotic cells.;The last application presented in the current dissertation is gene delivery. A MSN based gene transfection system was developed, where second generation (G2) PAMAMs were covalently attached to the surface of MSN. The G2-PAMAM-capped MSN material (G2-MSN) was able to complex with a plasmid DNA (pEGFP-C1). The gene transfection efficacy, uptake mechanism, and biocompatibility of the G2-MSN system with various cell types were investigated
Facilitated Release of Doxorubicin from Biodegradable Mesoporous Silica Nanoparticles
Cervical cancer is one of the most common causes of cancer death for women in the United States. The current treatment with chemotherapy drugs has significant side effects and may cause harm to healthy cells rather than cancer cells. In order to combat the potential side effects, nanoparticles composed of mesoporous silica were created to house the chemotherapy drug doxorubicin (DOX). The silica network contains the drug, and a pH study was conducted to determine the conditions for the nanoparticle to disperse the drug. The introduction of disulfide bonds within the nanoparticle created a framework to efficiently release 97% of DOX in acidic environments and 40% release in neutral environments. The denotation of acidic versus neutral environments was important as cancer cells are typically acidic. The chemistry was proved with the incubation of the loaded nanoparticle into HeLa cells for a cytotoxicity report and confocal imaging. The use of the framework for the anticancer drug was shown to be effective for the killing of cancerous cells
Spatiotemporal patterns and predictability of cyberattacks
A relatively unexplored issue in cybersecurity science and engineering is
whether there exist intrinsic patterns of cyberattacks. Conventional wisdom
favors absence of such patterns due to the overwhelming complexity of the
modern cyberspace. Surprisingly, through a detailed analysis of an extensive
data set that records the time-dependent frequencies of attacks over a
relatively wide range of consecutive IP addresses, we successfully uncover
intrinsic spatiotemporal patterns underlying cyberattacks, where the term
"spatio" refers to the IP address space. In particular, we focus on analyzing
{\em macroscopic} properties of the attack traffic flows and identify two main
patterns with distinct spatiotemporal characteristics: deterministic and
stochastic. Strikingly, there are very few sets of major attackers committing
almost all the attacks, since their attack "fingerprints" and target selection
scheme can be unequivocally identified according to the very limited number of
unique spatiotemporal characteristics, each of which only exists on a
consecutive IP region and differs significantly from the others. We utilize a
number of quantitative measures, including the flux-fluctuation law, the Markov
state transition probability matrix, and predictability measures, to
characterize the attack patterns in a comprehensive manner. A general finding
is that the attack patterns possess high degrees of predictability, potentially
paving the way to anticipating and, consequently, mitigating or even preventing
large-scale cyberattacks using macroscopic approaches
Rapid identification and application of Lactobacillus plantarum, Lactobacillus paracasei and Lactobacillus pentosus using multiplex polymerase chain reaction and species-specific primers, targeting 16S ribosomal RNA and recA genes
Background and Objective: Various products on the market contain probiotics such as lactic acid bacteria, which are promoted with a wide range of benefits. Functionality of these products is linked to the specific strains, bacterial species and viable cell counts. This study aimed to assess conformity of targeted lactic acid bacterial species and viable cell counts in commercially available probiotic products with their labeling, ensuring efficacy of the products.
Material and Methods: Multiplex polymerase chain reaction technique was developed using specific primers to effectively differentiate lactic acid bacteria in probiotic products. Therefore, strains used in the products were targeted and relevant nucleotide sequence data were searched to select two sets of polymerase chain reaction primer pairs of L. pla-F/R and L. para-F/R, targeting 16S ribosomal RNA genes, and L .pen-F/R, targeting recA genes.
Results and Conclusion: The individual primer sets produced the expected target products that matched the labeling for the tested strains of Lactobacillus plantarum, Lactobacillus paracasei and Lactobacillus pentosus. Then, specificity assessing was carried out using multiplex primer sets for single strains, pairwise combinations and triple combinations of lactic acid bacteria. After verifying specificity for all the three strains under similar polymerase chain reaction conditions, sensitivity of the multiplex polymerase chain reaction was investigated by assessing various dilutions of the three lactic acid bacterial strains and commercially available probiotic products. These findings demonstrated potential uses of multiplex polymerase chain reaction in lactic acid bacterial detection techniques. In conclusion, specific primer sets can be used in multiplex polymerase chain reaction to rapidly and effectively detect lactic acid bacterial strains in commercial products.
Conflict of interest: The authors declare no conflict of interest.
Lactic acid bacteria (LAB) are addressed for their beneficial effects on the human gastrointestinal tract, enhan-cing overall immune health. There are recent interests in development of functional LAB products. Demands for the probiotic functional foods are rapidly increasing due to the increased consumer awareness of food effects on health [1]. Researchers have assessed potential uses of probiotics in dairy and nondairy products and their viabilities during storage [2]. They have concluded that the final product should contain a minimum of 106-107 viable cells per serving to benefit consumer health [2]. Their studies have shown that mislabeling of probiotic species is common in commercial products [3]. Lack of appropriate identification of the strains and false efficacy claims have led to confusion. Probiotic products available in the market often use mixed strains. Thus, there are needs to monitor conformities of the labeled bacterial species and viable cell counts.
Polymerase chain reaction (PCR) technique has success-fully been used to detect and differentiate viruses and bacteria in various foods [4]. Rapid and reliable nature of PCR provides a valuable tool for distinguishing closely related species within two groups of lactobacilli [5]. This technique has been used to rapidly identify Lactobacillus plantarum in kimchi [6]. Oligonucleotide primers have been developed from sequences between the 16S and 23S rRNA genes, enabling identification of various lactobacilli strains in dairy products and probiotics using PCR [7]. Further-more, strain-specific PCR can be used for the rapid identi-fication of lactobacilli isolated from food samples [8] and specific identification of ten common lactobacilli and bifi-dobacteria strains in fermented milks [9].
In this study, multiplex PCR method was developed to investigate applicability of molecular detection techniques for LAB using three sets of specific primers sourced from the literature. Moreover, 16S rRNA gene sequences were targeted to explore molecular LAB detection. By amplifying 16S rRNA and recA gene sequences through PCR, this study simultaneously detected three LAB strains as well as mixed LAB strains in the products. Sensitivity of detection was assessed to establish a simple, reliable rapid method appropriate for the effective identification of LAB strains in probiotic products.
Materials and Methods
2.1 Bacterial strains and culture conditions
The LAB strains were stored in a -80 °C freezer. Before the experiments, strains were activated twice using lactobacilli MRS broth (Difco, Detroit, MI, USA) supplemented with 0.05% (w/w) L-cysteine (Merck, Taipei, Taiwan). Strains were cultured under optimal growth conditions at 37 °C for 24 h. Reference strains were provided by the Bioresources Collection and Research Center (BCRC, Hsin-Chu, Tai-wan). The L. pentosus BCRC 17972 and 17973, L. plantar-um F7-1 and L. paracasei BCRC 12193, 12188, 12248 and 17002 were used in this study. Seven strains were used in the current study as well. The commercial probiotic product was purchased from Li-Fong, Tainan, Taiwan, for PCR detection. Each sachet of the bacterial powder contained a high level of viable probiotic cells with 5.0 × 1010 CFU.g-1. Specific strains included in the product were L. plantarum LP112, L. paracasei LPC188 and L. pentosus LPE588. ThIS study was carried out at Testing and Analysis Center for Food and Cosmetics, HungKuang University, Taichung City, Taiwan.
2.2 Genomic DNA preparation and polymerase chain reaction primers
Total chromosomal DNA of the LAB cells was extracted using Blood and Tissue Genomic DNA Extraction Miniprep System (Viogene, Taipei, Taiwan) based on the manuf-acturer’s instructions. Specific primer sequences for the LAB detection are shown in Table 1. Experiments were repeated thrice [10-12].
2.3 Polymerase chain reaction amplification
Method was carried out according to [13]. For each PCR cycle, denaturation, annealing and extension were carried out at 94 °C for 60 s, 57 °C for 60 s and 72 °C for 120 s, respectively. Final extension was carried out at 72 °C for 5 min.
2.4 Sensitivity of the polymerase chain reaction assay
A 24-h culture of the LAB strain was serially diluted 10-fold with sterile water. Purification of DNA was carried out as described in Section 2.2 [14].
2.5 Polymerase chain reaction detection in the commercial probiotic product
The probiotic product was purchased from Li-Fong, Tainan, Taiwan. After diluting the product to 108–106, 1 ml of the diluted sample was collected and DNA extraction was carried out. Then, 2 μl of the extracted DNA was used for multiplex PCR. Experiments were repeated thrice.
Results and Discussion
3.1 Multiplex Polymerase chain reaction
Figure 1 shows gel electrophoresis results of the multiplex PCR for DNA detection of individual LAB strains. Results demonstrated specificity of the three primer sets for their respective target genes in each strain. Small interferences were seen for L. plantarum F7-4 with no effects on amplification of other strains. Figure 1 shows gel electrophoresis results of multiplex PCR for DNA detection of two LAB strains. Results demonstrated that L.pla-F/R, L.pen-F/R and L.para-F/R primer sets could accurately amplify DNA from combination of two LAB strains. No nonspecific products were observed. Thus, 57°C was determined as the optimal annealing temperature for successful primer binding and DNA polymerase activity.
Gel electrophoresis results validated effectiveness and specificity of the multiplex PCR for identifying and discriminating various LAB strains. These findings demonstrated applicability of the developed primer sets and verified their suitability for use in multiplex PCR. The optimized annealing temperature ensured robust amplifica-tion without nonspecific amplification products.
3.2 Sensitivity assessment of the lactic acid bacterial strains using multiplex polymerase chain reaction
In Figure 2A, DNA extracted directly from the mixed cultures of three LAB strains (109, 108 and 107 cfu ml-1) are shown. At a concentration of 106 cfu ml-1, only L. paracasei (BCRC 12188) demonstrated amplifications in multiplex PCR, indicating that detection sensitivity of L. plantarum and L. pentosus was limited to 107 cfu ml-1. Figure 2B shows mixed cultures of the three LAB strains at similar concen-trations after preculture in MRS broth (37 °C, 24 h). Multi-plex PCR detected all the three strains at a sensitivity of 106 cfu ml-1, suggesting that L. plantarum and L. pentosus proliferated and could be detected. Figure 2 shows detection limits and proliferation capabilities of the three LAB strains using multiplex PCR. Results indicated that preculture of the mixed cultures in MRS broth improves detection sensitivity, enabling accurate identification of L. plantarum and L. pentosus strains at lower concentrations.
3.3 Preculture and mixed various concentrations of lactic acid bacterial strains using multiplex polymerase chain reaction
Figure 3A shows gel electrophoresis results of the multiplex PCR carried out on DNA samples extracted from a mixture of L. plantarum (109 cfu ml-1) with two other LAB strains after preculture for 24 h. In Lanes 5–8, amplification products of L. pentosus and L. paracasei diluted to 106 cfu ml-1 were less expressed. The PCR amplification products of 106 cfu ml-1 DNA could be observed. The L. pentosus BCRC 17973 demonstrated a bacterial count of ~109 cfu ml-1 after 24 h of cultivation, whereas L. paracasei BCRC 12188 showed increased bacterial count, suggesting that preculture could enhance detection rate of low-concen-tration bacterial strains. Figure 3B shows gel electrophoresis results of the multiplex PCR on DNA extracted from a mixture of L. plantarum (108 cfu ml-1) with two other LAB strains after preculture for 24 h. The PCR products in Lanes 1–16 indicated that all the three sets of species-specific primer pairs yielded the expected PCR products for various concentrations of the bacterial suspensions after 24 h of preculture. Figure 3 shows effectiveness of the multiplex PCR in detecting L. plantarum at various concentrations in a mixed culture with other LAB strains. Results highlighted effects of preculture on enhancing assay detection rate and sensitivity.
3.4 Polymerase chain reaction detection of the commercial probiotic product
Figure 4 demonstrates results of the multiplex PCR. Whether directly detected or precultured, the three LAB strains provided DNA amplification products from 107 to 109 cfu ml-1. It was suggested that 107 cfu ml-1 was the detection limit of the product (not detected when diluted to 106 cfu ml-1).
The PCR-based species identification is a critical highly valuable tool for detecting and identifying bacteria. It offers advantages, including time efficiency and reliability in microbial identification. Compared to traditional methods such as culture-based techniques, PCR can provide results in a relatively short time. It eliminates the need of time-consuming cultivation of bacteria, allowing for further rapid identification and subsequent decision-making processes. In industrial uses, species identification is critical in selecting and developing bacterial strains appropriate for specific purposes.
Whether in food, agriculture or biotechnology industries, PCR-based species identification enables researchers to screen and select the most appropriate bacterial species for desired characteristics and functions.
The 16S rRNA gene in the ribosomal RNA of prokaryotes is the best molecular marker for bacterial evolutionary analysis. This is due to several gene characteristics, including its presence across various species, abundance, sufficient sequence length and presence of conserved and variable loci. The 16S rRNA gene is widely used in identifying lactobacilli and a commonly rapid technique for bacterial classification and identification in dairy products. Caro et al. reported that partial sequencing of 16S rRNA genes is often used for lactobacilli identification [15]. The RecA protein is a DNA recombinase that plays critical roles in DNA repair and recombination processes of bacteria. It is encoded by the recA gene in the genomes of various prokaryotic microorganisms. The recA gene and its corresponding protein have extensively been studied and used in various research, including evolutionary analysis, phylogenetic studies and identification of bacterial strains. Conserved nature and functional importance of the RecA protein make it a valuable molecular marker for understanding genetic relatedness and evolutionary relationships within bacteria. By comparing the recA gene sequences of various bacterial strains, researchers can have insights into their genetic diversity, evolutionary history and phylogenetic classification.
Lu et al. developed a multiplex PCR method that could effectively be used in key vaginal microbiota evaluation in women with bacterial vaginosis [16]. You et al. used multiplex PCR to detect six species of L. acidophilus group [17]. Petri et al. used multiplex PCR for the rapid identification of wine-associated LAB [18]. Settanni et al. introduced a method and reported its state-of-the-art uses for microbial identification in foods and beverages [19]. Sciancalepore et al. described use of a simple, low-cost, rapid sensitive method based on droplet-based multiplex PCR directly on food matrices for the simultaneous detection of bacterial genes involved in biogenic amine synthesis [20]. Specific primers were designed based on similar sequences and multiplex PCR was optimized for the simultaneous identification of L. plantarum, L. pentosus and L. paraplantarum [21]. Sul et al. developed a multiplex PCR to detect Lactobacillus and Bifidobacterium spp. in commercial probiotic products [13].
Previously, Gram-negative broth enrichment was used in studies, followed by immunomagnetic separation multiplex PCR to enable the simultaneous detection of Salmonella spp. and enterohemorrhagic Escherichia coli in food samples, irrespective of their significant discrepancy in cell counts [22]. A PCR primer set derived from the seque-nce in 16S to 23S internal transcribed spacer (ITS) region was also developed for the specific detection of B. adoles-centis in probiotics. This primer set included potentials for inspecting dairy food and environmental samples [14]. To ensure food safety during direct vat inoculation of Paocai, a propidium monoazide-based quantitative PCR method was developed to quantify L. plantarum NCU116 fermentation starter, as well as Saccharomyces spp. and potentially present pathogenic bacteria [23]. Plate counting was carried out and demonstrated similar results to quantitative PCR analysis, indicating appropriateness and effectiveness of absolute quantitative PCR for rapidly detecting microbial composition in the Paocai system [23]. In a recent study, surveillance of Lactobacillus bacteremia was carried out using biochemical and conventional-PCR assays. However, these methods could not provide target quantification and might lead to false-positive results [24]. To address this limitation, a L. rhamnosus-specific quantitative PCR assay was developed. This assay delivers accurate and reproduci-ble results, leveraging specificity of a TaqMan probe, targ-eting unique 16S rDNA sequences of L. rhamnosus [24].
Conclusion
In summary, the three sets of PCR primer combinations, targeting various LAB strains, demonstrated specificity and generated the expected amplicons during PCR amplifica-tion. Use of multiplex PCR in LAB genomic detection showed potentials and was appropriate for detecting various species in food products. Multiplex PCR decreased experimental cost and time, eliminating the need of time-consuming sequencing processes. Therefore, this method is expected to contribute to the reliability of probiotic labeling systems by facilitating strain identification. This molecular technique offers a valuable tool for quality control, product development and microbial monitoring of probiotic strains in various fields.
Conflict of Interest
The authors report no conflict of interest.
Authors Contributions
Conceptualization, TCC; methodology, LZY; data curation, LZY; writing, TCC.
References
Latif A, Shehzad A, Niazi S, Zahid A, Ashraf W, Iqbal MW, Rehman A, Riaz T, Aadil RM, Khan IM, Özogul F, Rocha JM, Esatbeyoglu T, Korma SA. Probiotics: mechanism of action, health benefits and their application in food industries. Front Microbiol. 2023; 14: 1216674.
https://doi.org/10.3389/fmicb.2023.1216674
Jena R, Choudhury PK. Bifidobacteria in Fermented Dairy Foods: A Health Beneficial Outlook. Probiotics Antimicrob Proteins. 2023; Online ahead of print.
https://doi.org/10.1007/s12602-023-10189-w
Yeung PSM, Sanders ME, Kitts CL, Cano R, Tong PS. Species-specific identification of commercial probiotic strains. J Dairy Sci. 2002; 85 (5):1039-1051.
https://doi.org/10.3168/jds.S0022-0302(02)74164-7Â
Pyar H, Liong MT, Masazurah AR, Lew LC, Peh KK. Study of genotypic characteristics of probiotics Lactobacillus using PCR. Asian Pac J Trop Dis. 2014; 4 (3): 225.
https://doi.org/10.1016/S2222-1808(14)60515-6
Berthier F, Ehrlich S D. Rapid species identification within two groups of closely related lactobacilli using PCR primers that target the 16S/23S rRNA spacer region. FEMS Microbiol Lett. 1998; 161 (1): 97-106.
https://doi.org/10.1111/j.1574-6968.1998.tb12934.x
Kim TW, Min SG, Choi DH, Jo JS, Kim HY. Rapid identification of Lactobacillus plantarum in kimchi using polymerase chain reaction. J Microbiol Biotech. 2000;10(6): 881-884.
Tilsala-Timisjaervi A, Alatossava T. Development of oligonucleotide primers from the 16S-23S rRNA intergenic sequences for identifying different dairy and probiotic lactic acid bacteria by PCR. Int J Food Microbiol. 1997; 35 (1): 49-56.
https://doi.org/10.1016/S0168-1605(97)88066-X
Rossetti L, Giraffa G. Rapid identification of dairy lactic acid bacteria by M13-generated, RAPD-PCR fingerprint databases. J Microbiol Methods. 2005; 63 (2): 135-144.
https://doi.org/10.1016/j.mimet.2005.03.001
Lu W, Kong W, Yang P, Kong J. A one-step PCR-based method for specific identification of 10 common lactic acid bacteria and Bifidobacterium in fermented milk. Int Dairy J. 2015; 41, 7-12.
https://doi.org/10.1016/j.idairyj.2014.08.020
Tajabadi N, Mardan M, Saari N, Mustafa S, Bahreini R, Manap MY. Identification of Lactobacillus plantarum, Lactobacillus pentosus and Lactobacillus fermentum from honey stomach of honeybee. Braz J Microbiol. 2014; 44(3): 717-722.
https://doi.org/10.1590/s1517-83822013000300008
Sheu SJ, Hwang WZ, Chen HC, Chiang YC, Tsen HY. Development and use of tuf gene-based primers for the multiplex PCR detection of Lactobacillus acidophilus, Lactobacillus casei group, Lactobacillus delbrueckii and Bifidobacterium longum in commercial dairy products. J Food Prot. 2009; 72 (1): 93-100.
https://doi.org/10.4315/0362-028x-72.1.93
Song Y, Kato N, Liu C, Matsumiya Y, Kato H, Watanabe K. Rapid identification of 11 human intestinal Lactobacillus species by multiplex PCR assays using group- and species-specific primers derived from the 16S-23S rRNA intergenic spacer region and its flanking 23S rRNA. FEMS Microbiol Lett. 2000;187(2):167-173.
https://doi.org/10.1111/j.1574-6968.2000.tb09155.x
Sul, YS, Kim HJ, Kim TW, Kim HY. Rapid identification of Lactobacillus and Bifidobacterium in probiotic products using multiplex PCR. J Microbiol Biotechnol. 2007; 17 (3):490-495.
Tsai CC, Lai CH, Yu B, Tsen HY. Use of specific primers based on the 16S-23S internal transcribed spacer (ITS) region for the screening Bifidobacterium adolescentis in yogurt products and human stool samples. Anaerobe. 2008;14(4): 219-223.
https://doi.org/10.1016/j.anaerobe.2008.05.001
Caro I, Bécares G, Fuentes L, Garcia-Armesto MR, Rúa J, Castro JM, Quinto EJ, Mateo J. Evaluation of three PCR primers based on the 16S rRNA gene for the identification of lactic acid bacteria from dairy origin. CyTA-J Food. 2015; 13 (2): 181-187.
https://doi.org/10.1080/19476337.2014.934297
Lu S, Li Z, Chen X, Chen F, Yao H, Sun X, Cheng Y, Wang L, Dai P. Vaginal microbiota molecular profiling and diagnostic performance of artificial intelligence-assisted multiplex PCR testing in women with bacterial vaginosis: a single-center experience. Front Cell Infect Microbiol. 2024; 14:1377225.
https://doi.org/10.3389/fcimb.2024.1377225Â
You I, Kim EB. Genome-based species-specific primers for rapid identification of six species of Lactobacillus acidophilus group using multiplex PCR. PLoS One. 2020; 15(3): e0230550.
https://doi.org/10.1371/journal.pone.0230550Â
Petri A, Pfannebecker J, Fröhlich J, König H. Fast identification of wine related lactic acid bacteria by multiplex PCR. Food Microbiol. 2013; 33 (1): 48-54.
https://doi.org/10.1016/j.fm.2012.08.011
Settanni L, Corsetti A. The use of multiplex PCR to detect and differentiate food- and beverage-associated micro-organisms: A review. J Microbiol Methods. 2007; 69 (1): 1-22.
https://doi.org/10.1016/j.mimet.2006.12.008Â
Sciancalepore AG, Mele E, Arcadio V, Reddavide F, Grieco F, Spano G, Lucas P, Mita G, Pisignano D. Microdroplet-based multiplex PCR on chip to detect foodborne bacteria producing biogenic amines. Food Microbiol. 2013; 35 (1): 10-14.
https://doi.org/10.1016/j.fm.2013.02.010Â
Torriani S, Felis GE, Dellaglio F. Differentiation of Lactobacillus plantarum, L. pentosus and L. paraplantarum by recA gene sequence analysis and multiplex PCR assay with recA gene-derived primers. Appl Environ Microbiol. 2001; 67 (8): 3450-3454.
https://doi.org/10.1128/AEM.67.8.3450-3454.2001
Tsai CC, Hsih HY, Tsai CH, Tsen HY. Use of gram-negative medium and immunomagnetic separative method followed by multiplex polymerase chain reaction for th
The strip entropy approximation of Markov shifts on trees
The strip entropy is studied in this article. We prove that the strip entropy
approximation is valid for every ray of a golden-mean tree. This result extends
the previous result of [Petersen-Salama, Discrete \& Continuous Dynamical
Systems, 2020] on the conventional 2-tree. Lastly, we prove that the strip
entropy approximation is valid for eventually periodic rays of a class of
Markov-Cayley trees
- …