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Posted at the Institutional Resources for Unique Collection and Academic Archives at Tokyo Dental College, Available from http://ir.tdc.ac.jp/

Title

Influence of chewing on expression level of human

beta-defensin 2 and secretory immunoglobulin A in

the epithelium

Author(s)

Alternative

Ishida, A; Wada, T; Ogami, K; Ueda, T; Sakurai, K

Journal

Journal of prosthodontic research, 63(2): 162-166

URL

http://hdl.handle.net/10130/5102

Right

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Original Article

Influence of chewing on expression level of human beta-defensin 2 and Secretory

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Abstract

Purpose: This study aimed to clarify the influence of chewing on human β-defensin 2 (hBD-2)

and secretory immunoglobulin A (SIgA) expression levels.

Methods: We included 15 healthy males with no missing teeth (mean age, 25.5 ± 2.5 years).

Subjects were instructed to chew a piece of gum for 30 min. Saliva and skin-extraction

samples were collected before and after chewing for 15 and 30 min. hBD-2 and SIgA

concentrations in the samples were determined using enzyme-linked immunosorbent assay

(ELISA). hBD-2 and SIgA expression levels before and after chewing were analyzed using the

Mann–Whitney U test, following the Friedman test. The significance level was 0.05.

Results: The hBD-2 level in skin-extraction samples was significantly different before (99.4 ±

17.3 pg/mL) and after chewing for 30 min (142 ± 23.0 pg/mL). The SIgA level in skin-extraction

samples was also significantly different before (2.39 ± 0.25 μg/mL) and after chewing for 30 min (3.61 ± 0.33 μg/mL). No significant difference was noted in either hBD-2 or SIgA secretion rate in saliva between before and after chewing.

Conclusions: Chewing gum for 30 min increased hBD-2 and SIgA expression levels in skin.

Moreover, chewing gum could influence the secretion pattern of these two biomolecules on

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1. Introduction

Gastrointestinal infections, respiratory infections including aspiration pneumonia, and

skin infections readily develop and exacerbate because of age-related changes in immune

function [1,2]. Severe infections are difficult to treat and impair quality of life (QOL) [3,4].

Infections are primarily treated with antimicrobial drugs; this often leads to problems such as

the emergence of resistant bacteria and onset of adverse drug reactions [5]. Therefore,

activating immune function and increasing resistance to infections is important for

maintenance of QOL.

Human β-defensin 2 (hBD-2) and secretory immunoglobulin A (SIgA) are immune components extensively expressed in mucosa and epidermis, which are responsible for the

first line of host defense. hBD-2 is produced by epithelial cells [6]; it kills bacteria, fungi, and

protozoa [7], and inhibits the adherence of these pathogens to host cells. SIgA is produced in

glandular tissues, such as the salivary and sweat glands, and then secreted on the epithelial

surface [8-10].

Some previous studies have shown that whole-body physical activities, such as

stretching and walking, increase hBD-2 and SIgA expression levels [11-13]. However, some

people find it difficult to exercise because of physical disabilities. Thus, we focused on chewing

as a relatively safe physical activity that can be performed on a daily basis. hBD-2 expression

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which is influenced by the autonomic nervous system. Moreover, SIgA secretion from

glandular tissue, including saliva and sweat glands, is known to be directly regulated by

autonomic nervous activity.

Ohta et al. clarified that chewing changes the activity of the autonomic nervous system

[14]. Such changes influence systemic endocrine kinetics; hence, chewing may activate

immune function in a manner similar to that of whole-body physical activity. Elucidating the

influence of chewing on immune function may lead to the establishment of a method to safely

increase immune function. This also gives may emphasize the significance of lifelong oral

intake and chewing.

The aim of this study was to evaluate the influence of chewing on hBD-2 and SIgA

expression levels before and after chewing. Thus, we established a null hypothesis that

chewing gum does not increase the expression levels of hBD-2 or SIgA in the skin and saliva.

2. Materials and methods

2.1. Participants

We included 15 male students aged 20–29 years with no missing teeth from ********. Factors known to influence immune function include systemic infectious diseases, various

drugs, and hormone balance [15]. Therefore, subjects with inflammatory disease, those

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were included because sex hormone-related variation in immune function is lesser in males

than in females. Mental stress is known to influence immune function through changes in the

activity of the autonomic nervous system [16]. To eliminate the influence of stress conditions

on the experimental results, subjects’ anxiety states were evaluated immediately before the experiment using the State-Trait Anxiety Index Form X (STAI). In accordance with the manual

of STAI-Form X Japanese version, subjects with STAI score ≥ 42 were regarded as exhibiting stress; thus, they were excluded.

The study was explained to all subjects and written consent to participate was obtained

before the study. The study was performed after receiving approval from the ******** Ethics

Committee (approval number: 627).

2.2 Experimental Protocol

The time of the experiment was designated as 5:00 pm, with consideration of salivary

flow and diurnal variation of cortisol secretion from the adrenal cortex [17,18]. Moreover, to

eliminate dietary influence, we instructed all participants not to eat and/or drink, beginning 2

hours before the experiment. To achieve uniformity of the experimental environment among all

subjects, the conditions of the experimental room were controlled (temperature, 24 ± 2°C;

relative humidity, 40%). The subjects were instructed to rest for 20 min after entering the

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Then, the subjects freely chewed a tasteless chewing gum (Soft type 1.0 g; Lotte, Tokyo,

Japan). According to survey on time use and leisure activities by Japan Ministry of Internal

Affairs and Communications (2010), Japanese people spent time approximately 30 minutes

for a meal. Hence, we determined the chewing time for 30 minutes. The second sample was

collected after chewing for 15 min and the third after chewing for another 15 min.

2.3. Sample Collection and Analysis

We focused on saliva and skin-extraction samples, as they can be noninvasively

collected. The saliva samples were collected into centrifuge tubes (WATSON 50m Centrifuge

Tubel; Fukae-kasei, Tokyo, Japan) by 1-min salivation and immediately stored at −80°C until measurement. Skin-extraction samples were collected following the method reported by Eda

et al. [12]. A polypropylene tube cut into a ring with a 2-cm height was placed on the medial

skin of the forearm, and 1 mL of the solvent (150 mM NaCl, 50 mM Tris, 1% Tergitol, 0.5%

Deoxichol acid, 0.1% sodium dodecylsulfate) was added (Fig. 1). Using a microtube

homogenizer (23M; As One, Tokyo, Japan), the solvent was stirred at 9000 rpm for 1 min and

collected using a pipette. The collected sample was transferred into a microtube and stored at

−80°C until analysis. For quantitation of hBD-2 and SIgA, ELISA kits (hBD-2: β-Defensin 2 ELISA Kit EK-072-37; Phoenix Pharmaceuticals Inc., Burlingame, CA, USA; SIgA: Secretory

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hBD-2 and SIgA secretion rates (hBD-2: pg・min-1, SIgA: μg・min-1) were calculated by the

product of concentration (hBD-2: pg/mL, SIgA: μg/mL) and saliva secretion rate (mL・min-1). The concentration in skin-extraction samples and secretion rate in saliva samples were treated

as the expression levels. Because sampling and hBD-2 and SIgA quantitation methods have

been established for saliva and skin-extraction samples, sample collection and analysis were

considered to be accurately performed in the present study.

The maximum voluntary occlusal force was measured using a pressure-sensor film

(Dental Prescale Type R 50H; GC, Tokyo, Japan). A horseshoe-shaped pressure-sensitive film

was placed on the dental arch, and the subject was instructed to clench it with maximum force

for 3 seconds. The pressure-sensitive film after measurement was analyzed using an occlusal

force measurement system (Occlusor FPD-707; GC).

The number of chewing cycles was recorded using a Holter electromyograph (Muscle

Tester ME3000P; MegaElectronics Kuopio, Finland). Bipolar surface electrodes (Blue Sensor

P-00-S; Medicotest, Olstykke, Denmark) were attached to the bulged region of bilateral

masseter muscles; muscle activity was then measured for 30 min, and chewing frequency was

calculated.

2.4 Statistical Analysis

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and SIgA expression levels before and after chewing were analyzed using the Mann–Whitney U test after the Friedman test. The P value was corrected using the Bonferroni method.

Correlations between changes in the hBD-2 and SIgA expression levels after chewing for 30

min in each sample, as well as the maximum occlusal force and chewing frequency, were

analyzed using Spearman’s rank correlation coefficient. The significance level was set at 0.05. Statistical analysis was performed using SPSS statistics ver. 22 (International Business

Machines Corporation, Chicago, IL, USA).

3. Results

hBD-2 expression levels in the saliva samples before chewing, after chewing for 15 min,

and after chewing for 30 min were 400 ± 62 pg・min−1 (mean ± SD), 346 ± 67 pg・min−1, and

290 ± 72 pg・min−1, respectively; hBD-2 expression levels in the skin-extraction samples were

99.4 ± 17.3 pg/mL, 109 ± 20 pg/mL, and 143 ± 23 pg/mL, respectively. SIgA expression levels

in the saliva samples before chewing, after chewing for 15 min, and after chewing for 30 min

were 228 ± 51 μg・min−1, 153 ± 24 μg・min−1, and 191 ± 22 μg・min−1, respectively; SIgA

expression levels in the skin-extraction samples were 2.39 ± 0.25 μg/mL, 3.23 ± 0.25 μg/mL, and 3.61 ± 0.33 μg/mL, respectively. A significant difference was observed in the hBD-2 and SIgA expression levels in the skin-extraction samples before and after chewing for 30 min. No

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chewing (Figs. 2 and 3).

On correlation analysis, no correlation was noted between the maximum occlusal force

or number of chewing cycles and changes in the SIgA and hBD-2 levels in either sample type

(Table 1).

4. Discussion

To clarify the influence of chewing on immune function, we investigated the relationships

between chewing and hBD-2 and SIgA expression levels. Significant differences were noted in

the skin hBD-2 and SIgA expression levels before and after chewing, supporting our

hypothesis. The sweat glands are responsible for SIgA secretion in the skin. The sweating

center of the sweat gland communicates with the autonomic nervous system in the

hypothalamus in the brain, and is influenced by sympathetic nervous activity [19]. Stimulation

by gum chewing influences the activities of the sympathetic and parasympathetic nervous

systems [8,20]; furthermore, it changes the activity of the hypothalamic–pituitary–adrenal axis (HPA axis), which controls endocrine kinetics [21,22]. Therefore, stimulation by gum chewing

may have changed the secretory kinetics of the sweat gland by changing the activity of the

sympathetic nervous system, and may have increased SIgA in the skin. Skin epithelial cells

are continuously stimulated by pathogenic microorganisms, endotoxins, and inflammatory

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communicated to the nucleus through the cell surface Toll-like receptors (TLRs) and result in

the production of hBD-2 mRNA [23]. This mechanism is regulated by the adrenocorticosteroid

hormone (coltizol), secreted as a result of the activity of the HPA axis. Coltizol inhibits hBD-2

production by inhibiting signal transduction from TLR. An inverse correlation between salivary

hBD-2 and cortisol levels has been reported [24]. In this study, cortisol secretion before and

after chewing was not evaluated. However, we have previously shown that chewing gum

reduces cortisol levels in saliva [25]. Therefore, in the present study, chewing gum may have

caused reduction of cortisol and enhancement of hBD-2 production in epithelial cells.

In addition, we previously conducted a pilot study to evaluate whether length of time in

the experimental room influences hBD-2 and SIgA expression levels. We found that the same

experimental room environment and length of time as used in the present study did not

influence hBD-2 and SIgA expression levels. Thus, we considered these immune components

to not be influenced by the experimental environment in the present study.

No significant difference was noted in either immune component in the saliva; however,

this may be because of the timing of sampling. In the salivary glands, plasma cells under the

basement membrane produce dimer IgA (dIgA). When the autonomic nervous system is

activated, polymeric immunoglobulin receptor (pIgR) expression on the surface of glandular

epithelial cells is promoted. Then, dIgA under the basement membrane binds to pIgR and is

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changes in autonomic nervous activity increased in 15–30 sec after the initiation of chewing [28]; the rate of SIgA secretion into saliva then decreased. This suggested that dIgA under the

basement membrane was washed out by brief chewing. The absence of a difference after

chewing for 15 and 30 min in this study may have been because IgA in the salivary gland was

washed out, as in the previous study. This might explain why there were no significant

differences in SIgA secretion into saliva. It may take time to produce IgA after washing out;

moreover, the salivary SIgA level may significantly increase by extending the gum chewing

time and observing the course after completion of chewing for a prolonged time. Salivary

hBD-2 did not increase after chewing for 30 min in any subject. HBD-2 production begins in

response to stimulation with chewing gum, unlike SIgA production, suggesting that more time

is required for hBD-2 to pass through the stratified squamous epithelium in the mouth and

transfer into the saliva. Furthermore, the hBD-2 expression level is influenced by the degree of

keratinization; the hBD-2 expression level is higher in keratinized epithelium than in ortho- or

non-keratinized epithelium [29]. The hBD-2 expression level is lower in the oral mucosa

epithelium than in the skin because of the degree of keratinization; therefore, the hBD-2

expression level may be less likely to be influenced by changes in the autonomic nervous

activity induced by stimulation with chewing gum. This might explain why there were no significant differences in hBD-2 concentration in the saliva samples.

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influencing the energy metabolic rate during chewing. The maximum occlusal force does not

directly reflect the energy metabolic rate during chewing. However, a correlation has been

noted between the maximum occlusal force and muscle mass of the masseter; the muscle

mass of the masseter may increase with increased maximum occlusal force [30]. We suspect

that energy metabolism indirectly reflects the influence on the energy metabolic rate because it

can be increased by activating more muscles. hBD-2 and SIgA expression levels are known to

be influenced by the intensity of physical activity [11,12,14]. This may occur because physical

activity influences immune function through a route mediated by the HPA axis, as well as by

promoting inflammatory cytokines, such as IL-1β and TNF-α [31]. However, physical activity such as mild exercise (the energy metabolic rate: 2.5–4.0 metabolic equivalents [METs]) reportedly does not induce production of inflammatory cytokines [32]; therefore, gum chewing

(approximate 1.5 METs) [33] was unlikely to induce production of inflammatory cytokines and

influence immune function. In physical activity with a relatively low energy metabolic rate,

physical activity-induced changes in autonomic nervous and HPA axis activities may influence

immune function. The results of this study suggest that in such physical activities, a difference

in the energy metabolic rate may have a small influence on changes in hBD-2 and SIgA.

There are some considerations for generalizing the experimental results. In this study,

only male participants were included because sex hormone-related variation in immune

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participants were required to continuously chew gum for 30 min. Because some participants

exhibited reduced salivary SIgA after chewing, there was a possibility that they experienced

stress during chewing in this experiment.

This study suggested that chewing influences immune function throughout the body.

Therefore, even in people for whom continuous physical activity has a risk, such as falls or

fracture, masticatory guidance and gum chewing may safely activate immune function.

Mechanisms similar to those of the SIgA secretion system in sweat glands and the hBD-2

expression system in epithelial tissue have been observed in the airway and intestine[34],

suggesting that sufficient chewing in daily life may lead to the prevention of infections in the

respiratory and gastrointestinal tracts.

5. Conclusion

Under the conditions of this study, 30-minute gum chewing increases the SIgA and

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6. References

[1] Weiskopf D, Weinberger B, Grubeck-Loebenstein B. The aging of the immune system.

Transpl Int 2009;22:1041-50.

[2] Kish TD, Chang MH, Fung HB. Treatment of skin and soft tissue infections in the elderly:

a review. Am J Geriatr Pharmacother 2010;8:485-513.

[3] Linder JA, Singer DE. Health-related quality of life of adults with upper respiratory tract

infections. J Gen Intern Med 2003;18:802-7.

[4] Leibovici L. Long-term consequences of severe infections. Clin Microbiol Infect

2013;19:510-2.

[5] Meyers BR, Wilkinson P. Clinical pharmacokinetics of antibacterial drugs in the elderly.

Implications for selection and dosage. Clin Pharmacokinet 1989;17:385-95.

[6] Liu AY, Destoumieux D, Wong AV, Park CH, Valore EV, Liu L, et al. Human

beta-defensin-2 production in keratinocytes is regulated by interleukin-1, bacteria, and the

state of differentiation. J Invest Dermatol 2002;118:275-81.

[7] Ganz T, Lehrer RI. Defensins. Curr Opin Immunol 1994;6:584-9.

[8] Corthésy B. Role of secretory IgA in infection and maintenance of homeostasis.

Autoimmun Rev 2013;12:661-5.

[9] Okada T, Konishi H, Ito M, Nagura H, Asai J. Identification of secretory immunoglobulin A

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[10] Carpenter GH, Proctor GB, Ebersole LE, Garrett JR. Secretion of IgA by rat parotid and

submandibular cells in response to autonomimetic stimulation in vitro. Int Immunopharmacol

2004;4:1005-14.

[11] Shimizu K, Kimura F, Akimoto T, Akama T, Otsuki T. Nishijima T, et al. Effects of exercise,

age and gender on salivary secretory immunoglobulin A in elderly individuals. Exerc Immunol

Rev 2007;13:55-66.

[12] Eda N, Shimizu K, Suzuki S, Lee E, Akama T. Effects of high-intensity endurance exercise on epidermal barriers against microbial invasion. J Sports Sci Med 2013;12:44-51.

[13] Eda N, Ito H, Shimizu K, Suzuki S, Lee E, Akama T. Yoga stretching for improving

salivary immune function and mental stress in middle-aged and older adults. J Women Aging

2018;30:227-241.

[14] Ohta M, Ueda T, Sakurai K. Effect of chewing or compressing food on autonomic

nervous activity in older adults. Gerodontology 2017;34:434-40.

[15] Giefing-Kröll C, Berger P, Lepperdinger G, Grubeck-Loebenstein B. How sex and age

affect immune responses, susceptibility to infections, and response to vaccination. Aging Cell

2015;14:309-21.

[16] Dhabhar FS. Psychological stress and immunoprotection versus immunopathology in

the skin. Clin Dermatol 2013;31:18-30.

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1972;220:529-45.

[18] Purnell JQ, Kahn SE, Samuels MH, Brandon D, Loriaux DL, Brunzell JD. Enhanced

cortisol production rates, free cortisol, and 11beta-HSD-1 expression correlate with visceral fat

and insulin resistance in men: effect of weight loss. Am J Physiol Endocrinol Metab

2009;296:E351-7.

[19] Asahina M, Poudel A, Hirano S. Sweating on the palm and sole: physiological and

clinical relevance. Clin Auton Res 2015;25:153-9.

[20] Hasegawa Y, Sakagami J, Ono T, Hori K, Zhang M, Maeda Y. Circulatory response and

autonomic nervous activity during gum chewing. Eur J Oral Sci 2009;117:470-3.

[21] Tsigos C, Chrousos GP. Hypothalamic-pituitary-adrenal axis, neuroendocrine factors and

stress. J Psychosom Res 2002;53:865-71.

[22] Ono Y, Yamamoto T, Kubo KY, Onozuka M. Occlusion and brain function: mastication as

a prevention of cognitive dysfunction. J Oral Rehabil 2010;37:624-40.

[23] Vora P, Youdim A, Thomas LS, Fukata M, Tesfay SY, Lukasek K, et al. Beta-defensin-2

expression is regulated by TLR signaling in intestinal epithelial cells. J Immunol

2004;173:5398-405.

[24] Usui T, Yoshikawa T, Orita K, Ueda S, Katsura Y, Fujimoto S, et al. Changes in salivary

antimicrobial peptides, immunoglobulin A and cortisol after prolonged strenuous exercise. Eur

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[25] Tasaka A, Takeuchi K, Sasaki H, Yoshii T, Soeda R, Ueda T, et al. Influence of chewing

time on salivary stress markers. J Prosthodont Res 2014;58:48-54.

[26] Carpenter GH, Proctor GB, Anderson LC, Zhang XS, Garrett JR. Immunoglobulin A

secretion into saliva during dual sympathetic and parasympathetic nerve stimulation of rat

submandibular glands. Exp Physiol 2000;85:281-6.

[27] Proctor GB, Carpenter GH. Regulation of salivary gland function by autonomic nerves.

Auton Neurosci 2007;133:3-18.

[28] Proctor GB, Carpenter GH. Chewing stimulates secretion of human salivary secretory

immunoglobulin A. J Dent Res 2001;80:909-13.

[29] Abiko Y, Suraweera AK, Nishimura M, Arakawa T, Takuma T, Mizoguchi I, et al.

Differential expression of human beta-defensin 2 in keratinized and non-keratinized oral

epithelial lesions; immunohistochemistry and in situ hybridization. Virchows Arch

2001;438:248-53.

[30] Castelo PM, Pereira LJ, Bonjardim LR, Gavião MB. Changes in bite force, masticatory

muscle thickness, and facial morphology between primary and mixed dentition in preschool

children with normal occlusion. Ann Anat 2010;192:23-6.

[31] Ostrowski K, Rohde T, Asp S, Schjerling P, Pedersen BK. Pro- and anti-inflammatory

cytokine balance in strenuous exercise in humans. J Physiol 1999;515:287-91.

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salivary beta-defensin 2. Eur J Appl Physiol 2013;113:2621-7.

[33] Ainsworth BE, Haskell WL, Herrmann SD, Meckes N, Bassett DR Jr, Tudor-Locke C, et

al. 2011 Compendium of physical activities: a second update of codes and MET values. Med

Sci Sports Exerc 2011;43:1575-81.

[34] Chairatana P, Nolan EM. Defensins, lectins, mucins, and secretory immunoglobulin A:

microbe-binding biomolecules that contribute to mucosal immunity in the human gut. Crit Rev

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7. Figures and Table Legends

Figure 1. Sample collection from the medial side of the forearm. A polypropylene tube cut into

a ring was placed on the medial skin of the upper arm, and the solvent was added and stirred

using a microtube homogenizer.

Figure 2. HBD-2 levels in saliva and skin samples. hBD-2 secretion into saliva (a) and hBD-2

concentration in skin-extraction samples (b) before and after chewing. The Mann–Whitney U test was used after the Friedman test (α = 0.05, P value was adjusted by Bonferroni correction). *Significant differences, ◦Outlier.

Figure 3. SIgA levels in saliva and skin samples. SIgA secretion into saliva (a) and SIgA

concentration in skin-extraction samples (b) before and after chewing. The Mann–Whitney U test was used after the Friedman test (α = 0.05, P value was adjusted by Bonferroni correction). *Significant differences, ◦Outlier.

Table 1. Relationships between the maximum occlusal force and chewing frequency, and

changes in hBD-2 and SIgA.

To analyze correlations between maximum occlusal force, numbers of chewing cycles,

and changes in hBD-2 and SIgA during chewing, Spearman’s rank correlation coefficients were calculated between maximum occlusal force and chewing frequency, and changes in the

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Table 2. Summary of participants’ maximal voluntary occlusal forces and numbers of chewing

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Figure 1

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Figure 2

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Figure 3

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Table 1 Table 2 Immune components Sample type

Maximal voluntary occlusal force Number of chewing cycles Correlation coefficient (r) p-value Correlation coefficient (r) p-value

hBD-2 Saliva 0.11 0.70 0.26 0.36 Skin -0.21 0.45 0.33 0.23 SIgA Saliva 0.44 0.10 -0.14 0.63 Skin 0.24 0.38 0.14 0.62

Participant No. Maximal voluntary occlusal force (N) Number of chewing cycle

1 818.5 2217 2 443.4 1553 3 518.7 2291 4 672.4 1785 5 841.3 2581 6 473.3 1716 7 861.5 1880 8 336.8 2623 9 763.6 2014 10 836.4 2235 11 788.9 2071 12 660.2 2913 13 411.9 1734 14 650.9 1876 15 507.1 2194

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