Three-dimensional finite element analysis of traumatic mechanism of mandibular symphyseal fracture combined with bilateral intracapsular condylar fractures / 北京大学学报(医学版)

OBJECTIVE@#To analyze the biomechanical mechanism of mandibular symphyseal fracture combined with bilateral intracapsular condylar fractures using finite element analysis (FEA).@*METHODS@#Maxillofacial CT scans and temporomandibular joint (TMJ) MRI were performed on a young male with normal mandible, no wisdom teeth and no history of TMJ diseases. The three-dimensional finite element model of mandible was established by Mimics and ANSYS based on the CT and MRI data. The stress distributions of mandible with different angles of traumatic loads applied on the symphyseal region were analyzed. Besides, two models with or without disc, two working conditions in occlusal or non-occlusal status were established, respectively, and the differences of stress distribution between them were compared.@*RESULTS@#A three-dimensional finite element model of mandible including TMJ was established successfully with the geometry and mechanical properties to reproduce a normal mandibular structure. Following a blow to the mandibular symphysis with different angles, stress concentration areas were mainly located at condyle, anterior border of ramus and symphyseal region under all conditions. The maximum equivalent stress always appeared on condylar articular surface. As the angle between the external force and the horizontal plane gradually increased from 0° to 60°, the stress on the mandible gradually concentrated to symphysis and bilateral condyle. However, when the angle between the external force and the horizontal plane exceeded 60°, the stress tended to disperse to other parts of the mandible. Compared with the condition without simulating the disc, the stress distribution of articular surface and condylar neck decreased significantly when the disc was present. Compared with non-occlusal status, the stress on the mandible in occlusal status mainly distributed on the occlusal surface, and no stress concentration was found in other parts of the mandible.@*CONCLUSION@#When the direction of external force is 60° from the horizontal plane, the stress distribution mainly concentrates on symphyseal region and bilateral condylar surface, which explains the occurrence of symphyseal fracture and intracapsular condylar fracture. The stress distribution of condyle (including articular surface and condylar neck) decreases significantly in the presence of arti-cular disc and in stable occlusal status when mandibular symphysis is under traumatic force.

Stress change of periodontal ligament of the anterior teeth at the stage of space closure in lingual appliances: a 3-dimensional finite element analysis / 北京大学学报(医学版)

OBJECTIVE@#To analyze the stress distribution in the periodontal ligament (PDL) under different loading conditions at the stage of space closure by 3D finite element model of customized lingual appliances.@*METHODS@#The 3D finite element model was used in ANSYS 11.0 to analyze the stress distribution in the PDL under the following loading conditions: (1) buccal sliding mechanics (0.75 N,1.00 N,1.50 N), (2) palatal sliding mechanics (0.75 N,1.00 N,1.50 N), (3) palatal-buccal combined sliding mechanics (buccal 1.00 N + palatal 0.50 N, buccal 0.75 N + palatal 0.75 N, buccal 0.50 N+ palatal 1.00 N). The maximum principal stress, minimum principal stress and von Mises stress were evaluated.@*RESULTS@#(1) buccal sliding mechanics(0.75 N,1.00 N,1.50 N): maximum principal stress: at the initial of loading, maximum principal stress, which was the compressed stress, distributed in labial PDL of cervix of lateral incisor, and palatal distal PDL of cervix of canine. With increasing loa-ding, the magnitude and range of the stress was increased. Minimum principal stress: at the initial of loading, minimum principal stress which was tonsil stress, distributed in palatal PDL of cervix of lateral incisor and mesial PDL of cervix of canine. With increasing loading, the magnitude and range of minimum principal stress was increased. The area of minimum principal stress appeared in distal and mesial PDL of cervix of central incisor. von Mises stress:it distributed in labial and palatal PDL of cervix of lateral incisor and distal PDL of cervix of canine initially. With increasing loading, the magnitude and range of stress was increased towards the direction of root. Finally, there was stress concentration area at mesial PDL of cervix of canine. (2) palatal sliding mechanics(0.75 N,1.00 N,1.50 N): maximum principal stress: at the initial of loading, maximum principal stress which was the compressed stress, distributed in palatal and distal PDL of cervix of canine, and distal-buccal and palatal PDL of cervix of lateral incisor. With increasing loading, the magnitude and range of the stress was increased. Minimum principal stress: at the initial of loading, minimum principal stress which was tonsil stress, distributed in distal-interproximal PDL of cervix of lateral incisor and mesial-interproximal PDL of cervix of canine. With increasing loading, the magnitude and range of the stress was increased.von Mises stress: von Mises stress distributed in palatal and interproximal PDL of cervix of canine. With increasing loading, the magnitude and range of stress was increased. Finally, von Mises stress distributing area appeared at distal-palatal PDL of cervix of canine. (3) palatal-buccal combined sliding mechanics: maximum principal stress: maximum principal stress still distributed in distal-palatal PDL of cervix of canine. Minimum principal stress: minimum principal stress distributed in palatal PDL of cervix of lateral incisor when buccal force was more than palatal force. As palatal force increased, the stress concentrating area transferred to mesial PDL of cervix of canine.von Mises stress: it was lower and more well-distributed in palatal-buccal combined sliding mechanics than palatal or buccal sliding mechanics.@*CONCLUSION@#Using buccal sliding mechanics,stress majorly distributed in PDL of lateral incisor and canine, and magnitude and range of stress increased with the increase of loading; Using palatal sliding mechanics, stress majorly distributed in PDL of canine, and magnitude and range of stress increased with the increase of loading; With palatal-buccal combined sliding mechanics, the maximum principal stress distributed in the distal PDL of canine. Minimum principal stress distributed in palatal PDL of cervix of lateral incisor when buccal force was more than palatal force. As palatal force was increasing, the minimum principal stress distributing area shifted to mesial PDL of cervix of canine. When using 1.00 N buccal force and 0.50 N palatal force, the von Mises stress distributed uniformly in PDL and minimal stress appeared.

[Risk factors of pulmonary arterial hypertension in patients with atrial septal defect at high altitude area].

OBJECTIVE: To analyze risk factors of pulmonary arterial hypertension (PAH) in patients with atrial septal defect (ASD) patients living at above 2000 m high altitude area. METHODS: We retrospectively analyzed the incidence of PAH in 526 ASD patients out of 1178 congenital heart disease patients who were hospitalized in Qinghai cardiovascular hospital between January 2007 to December 2009 and explored the risk factors including gander, age, altitude, defect size and nationalities for developing PAH in ASD patients using binary logistic regression. RESULTS: PAH prevalence was 68.4% (360/526) in ASD patients. The risk factors of developing PAH in these ASD patients were defect size (OR: 1.200, 95%CI: 1.156 - 1.246, P = 0.000), age (OR: 1.027, 95%CI: 1.003 - 1.052, P = 0.025) and altitude (OR: 1.389, 95%CI: 1.001 - 1.637, P = 0.043) while gender and nationality were not risk factors for PAH. The incidence of developing PAH increased with aging (P = 0.000). The standardized ratio of PAH at ≥ 3500 m was 74.8% which was significantly higher than that at 2000 - 2499 m altitude (66.2%, P = 0.005) and at 2500 - 3499 m altitude (66.9%, P = 0.005). CONCLUSIONS: The risk for developing PAH is high in patients living at high altitude area. The risk factors of developing PAH in ASD patients living at high altitude are defect size, age and altitude.

Individualized three-dimensional finite element model of facial soft tissue and preliminary application in orthodontics / 中华口腔医学杂志

<p><b>OBJECTIVE</b>To get individualized facial three-dimensional finite element (FE) model from transformation of a generic one to assist orthodontic analysis and prediction of treatment-related morphological change of facial soft tissue.</p><p><b>METHODS</b>A generic three-dimensional FE model of craniofacial soft and hard tissue was constructed based on a volunteer's spiral CT data. Seven pairs of main peri-oral muscles were constructed based on a combination of CT image and anatomical method. Individualized model could be obtained through transformation of the generic model based on selection of corresponding anatomical landmarks and radial basis functions (RBF) method. Validation was analyzed through superimposition of the transformed model and cone-beam CT (CBCT) reconstruction data. Pre- and post-treatment CBCT data of two patients were collected, which were superimposed to gain the amount of anterior teeth retraction and anterior alveolar surface remodeling that could be used as boundary condition. Different values of Poisson ratio ν and Young's modulus E were tested during simulation.</p><p><b>RESULTS</b>Average deviation was 0.47 mm and 0.75 mm in the soft and hard tissue respectively. It could be decreased to a range of +0.29 mm and -0.21 mm after a second transformation at the lip-mouth region. The best correspondence between simulation and post-treatment result was found with elastic properties of soft tissues defined as follows. Poisson ratio ν for skin, muscle and fat being set as 0.45 while Young's modulus being set as 90.0 kPa, 6.2 kPa and 2.0 kPa respectively.</p><p><b>CONCLUSIONS</b>Individualized three-dimensional facial FE model could be obtained through mathematical model transformation. With boundary condition defined according to treatment plan such FE model could be used to analyze the effect of orthodontic treatment on facial soft tissue.</p>

Patient-specific modeling of facial soft tissue based on radial basis functions transformations of a standard three-dimensional finite element model / 中华医学杂志(英文版)

<p><b>BACKGROUND</b>An important purpose of orthodontic treatment is to gain the harmonic soft tissue profile. This article describes a novel way to build patient-specific models of facial soft tissues by transforming a standard finite element (FE) model into one that has two stages: a first transformation and a second transformation, so as to evaluate the facial soft tissue changes after orthodontic treatment for individual patients.</p><p><b>METHODS</b>The radial basis functions (RBFs) interpolation method was used to transform the standard FE model into a patient-specific one based on landmark points. A combined strategy for selecting landmark points was developed in this study: manually for the first transformation and automatically for the second transformation. Four typical patients were chosen to validate the effectiveness of this transformation method.</p><p><b>RESULTS</b>The results showed good similarity between the transformed FE models and the computed tomography (CT) models. The absolute values of average deviations were in the range of 0.375 - 0.700 mm at the lip-mouth region after the first transformation, and they decreased to a range of 0.116 - 0.286 mm after the second transformation.</p><p><b>CONCLUSIONS</b>The modeling results show that the second transformation resulted in enhanced accuracy compared to the first transformation. Because of these results, a third transformation is usually not necessary.</p>

[Susceptibility of patients with congenital heart disease to pulmonary hypertension at a high altitude].

OBJECTIVE: To explore the incidence of congenital heart disease (CHD) complicated with pulmonary arterial hypertension (PAH) at an altitude of over 2000 meters above sea level. METHODS: We retrospectively analyzed the profiles of 688 PAH patients selected from 1178 CHD patients at our hospital during January 2007 to December 2009. The risk factors of CHD-PAH were analyzed by age, altitude, etiologies and nationalities. RESULTS: Among 1178 CHD patients, 688 (58.4%) suffered PAH. And 239 PAH cases (54.8%) were found in 436 male patients and 449 PAH cases (60.5%) in 742 female patients (χ(2) = 3.667, P = 0.058). The difference was not significant. The incidences of PAH in CHD patients varied at different altitudes (χ(2) = 10.507, P = 0.005). And 55.2%, 57.2% and 68.7% were found in < 2500 m, 2500 - 3500 m and > 3500 m groups respectively. And the > 3500 m group was statistically higher than those of the < 2500 m and 2500 - 3500 m groups (χ(2) = 9.924, P = 0.005) (χ(2) = 8.007, P = 0.005). The mean pulmonary arterial systolic pressure (PASP) of < 2500 m group was (58 ± 18) mm Hg, 2500 - 3500 m was (63 ± 17) mm Hg and > 3500 m was (64 ± 19) mm Hg. There were significant differences among them. In age composition of CHD-PAH, the highest ratio appeared before the age of 18 years old for PDA (patent ductus arteriosus)-PAH and VSD (ventricular septal defect)-PAH while that of ASD-PAH at the ages of 30 - 39 years old. There were not statistical differences among all nationalities (χ(2) = 5.404, P = 0.248). CONCLUSION: The incidence of CHD-PAH at a high altitude is significantly higher than those at a low altitude. Hypoxia may promote the occurrence and development of PAH in CHD patients.

Finite element study on the microdamage progression within bone / 医用生物力学

Objective To study the effects of mineral-collagen interfacial behavior on the microdamage progression within bone tissue. Methods Based on the finite element model, cohesive elements were introduced and the traction-separation law was used to simulate the role of ionic interactions, hydrogen bonds and van der waals forces. The effects of aforementioned interactions on the microdamage progression within bone were studied by the random field theory and probabilistic failure analysis. Results Strong interfaces (ionic interactions in both opening and sliding modes) between the mineral and collagen phases might encourage the formation of linear cracks in bone, whereas weak interfaces (van der Waals in opening mode and viscous shear in sliding mode) might facilitate the formation of diffuse damages. In addition, there existed a transitional interfacial bonding strength (hydrogen/van der Waals bonds) that governed the transition of microdamage accumulation from linear microcrack to diffuse damage.Conclusions The results from this study will help to understand the effects of mineral collagen interfacial behavior on microdamage accumulation in bone and further investigate the underlying mechanism of bone fracture due to osteoporosis or ageing.

Stress area of the mandibular alveolar mucosa under complete denture with linear occlusion at lateral excursion / 中华医学杂志(英文版)

<p><b>BACKGROUND</b>The rocking and instability of a loaded complete denture (CD) during lateral excursion reduce the bearing area under the denture base, causing localized high stress concentrations. This can lead to mucosal tenderness, ulceration, and alveolar bone resorption, and the linear occlusion design was to decrease the lateral force exerted on the denture and to ensure denture stability. But it is not known how the bearing areas of linear occlusal CDs (LOCDs) and anatomic occlusal CDs (AOCDs) differ. The purpose of this study was to analyze and compare the distributions of the high and low vertical stress-bearing areas in the mandibular alveolar mucosa under LOCDs and AOCDs at lateral excursion.</p><p><b>METHODS</b>Computerized tomography (CT) and finite element analysis were used to establish three-dimensional models of an edentulous maxilla and mandible with severe residual ridge resorption. These models were composed of maxillary and mandibular bone structure, mucosa, and the LOCD or AOCD. Lateral excursion movements of the mandible were simulated and the vertical stress-bearing areas in the mucosa under both mandibular CDs were analyzed using ANSYS 7.0.</p><p><b>RESULTS</b>On the working side, the high stress-bearing (-0.07 to -0.1 MPa) area under the LOCD during lateral excursion was smaller than that under the AOCD, while the medium stress-bearing (-0.03 to -0.07 MPa) area under the LOCD was 1.33-fold that under the AOCD. The medium stress-bearing area on the non-working side under the LOCD was 2.4-fold that under the AOCD. Therefore, the overall medium vertical stress-bearing area under the LOCD was 20% larger than that under the AOCD.</p><p><b>CONCLUSIONS</b>During lateral excursion, the medium vertical stress-bearing area under a mandibular LOCD was larger and the high vertical stress-bearing area was smaller than that under an AOCD. Thus, the vertical stress under the LOCD was distributed more evenly and over a wider area than that under the AOCD, thereby improving denture stability.</p>

[A cross-sectional study of congenital heart disease among children aged from 4 to 18 years at different altitudes in Qinghai province, China].

OBJECTIVE: The epidemiological characteristics of congenital heart disease (CHD) in children aged from 4 to 18 years were investigated in Qinghai province. METHODS: Altogether 288 066 children inhabiting at 6 prefectures and 3 counties were examined by the following three steps: pre-screening, re-examination and diagnosis with color Doppler. And the entity distribution was analyzed while the differences were compared by age, gender, altitudes and nationalities respectively. RESULTS: Altogether 1633 cases of CHD were discovered. The total prevalence of CHD was 5.71 per thousand. The prevalence of CHD was found to increase with the increase of altitude by 4.89 per thousand at the altitude of 2535 m, 5.71 per thousand at 3600 m, and 8.74 per thousand at 4200 m respectively. There were significant differences among different altitude (chi(2) = 54.696, P < 0.001). chi(2) trend analysis showed the increase with chi(2) = 41.826(P < 0.001). The total incidence of CHD in females was 6.95 per thousand, which was significantly higher than that in males with 4.54 per thousand (chi(2) = 73.79, P < 0.001). There were significant differences between males and females at the altitude of 3000 m (chi(2) = 84.733, P < 0.001) and 4000 m (chi(2) = 16.313, P < 0.001) except at the altitude of 2000 m (chi(2) = 0.807, P > 0.05). The prevalence of CHD in different age groups was statistically significant at the every altitude of 2000 m (chi(2) = 18.138, P < 0.001), 3000 m (chi(2) = 18.544, P < 0.001) and 4000 m (chi(2) = 27.535 P < 0.001). The prevalence of CHD was increasing with the increase of age groups at the altitude of 3000 m (chi(2) = 19.230, P < 0.001) and 4000 m (chi(2) = 26.894, P < 0.001) except at the altitude of 2000 m. Within the prevalence of CHD of different nationalities, there was a significant difference with chi(2) = 24.456 (P < 0.001). Within the constituent rate of CHD, the prevalence of atrial septal defect (ASD) was as high as 37.42%, followed by the prevalence of patent ductus arteriosus (PDA) as 28.47% and ventricular septal defect (VSD) as 26.01%. Regarding the four categories of CHD, the constituent rate varied at different altitudes. For example, the prevalence rate of ASD constituted 37% at the altitude of 2000 m and 3000 m, and that of PDA accounted for 46.36% at the altitude of 4200 m. CONCLUSION: The epidemiological characteristics of CHD in Qinghai children were possibly associated with altitude levels.

Stress distribution in alveolar bone around implants under implant supported overdenture with linear occlusion at lateral occlusion / 中华口腔医学杂志

<p><b>OBJECTIVE</b>To analyze stress distribution in alveolar bone around implants of implant supported overdentures (ISO) with linear occlusion and with anatomic occlusion at lateral mandibular position, and to justify the possibility of decreased injurious force around implants in ISO with linear occlusion.</p><p><b>METHODS</b>Computerized tomography scan and finite element analysis (FEA) were used to set up two 3-D FEA models of maxillae and mandible with severe residual ridge resorption. The mucosa, linear and anatomic occlusal ISO with bar attachments, and two implants inserted between mandibular foramina were also established in the models. With the condition of imitating the loading of masseter muscles, these models were loaded to simulate the stress distributions in alveolar bone around implants under ISO at lateral occlusion position.</p><p><b>RESULTS</b>At lateral occlusion, the stress distributions in alveolar bone around implants under ISO with anatomic occlusion were mainly on the lingual and distal sides of the working side implants. However, stress distributions under ISO with linear occlusion were on the distal sides of bilateral implants. Both the stress peaks of ISOs with linear occlusion and with the anatomic one appeared in the working side. In anatomic occlusion model, sigma(z): -6.47 MPa and 6.81 MPa, sigma(1): -4.20 MPa and 7.20 MPa (negative value: compressive stress, positive value: tensile stress); in linear occlusion model, sigma(z): -4.86 MPa and 3.04 MPa, sigma(1): -3.48 MPa and 5.33 MPa.</p><p><b>CONCLUSIONS</b>At lateral occlusion, when comparing the ISO with two different occlusion schemes, stress peak in alveolar bone around implants in the linear occlusion model was lower than that in the anatomic occlusion model at equal loading situation. Stress in the alveolar bone under ISO with linear occlusion distributed more evenly than that under ISO with anatomic occlusion.</p>

[Antiviral activity of a hammerhead ribozyme against HBV in HepG2.2.15 cells].

The carboxy terminal arginine rich domain of core protein is highly conserved among HBV subtypes, and plays important roles in the packaging of pregenomic RNA and viral replication. Therefore, the 3' highly conserved region of HBV C gene is an attractive target for antiviral therapy mediated by ribozymes. A hammerhead ribozyme RzC, targeting the above site, was designed; meanwhile, as controls, the disabled form of RzC, dRzC, was also designed. In order to investigate its antiviral effects in cultured cells, in vitro-transcribed ribozymeRzC, dRzC and ribozyme expression vectors pCRzC, pCdRzC were delivered, respectively, into HepG2.2.15 cells ( clonal cells derived from HepG2 cells that contain integrated HBV ayw DNA). The preliminary results demonstrated that in vitro-transcribed ribozyme RzC had weak inhibition on HBV replication, possibly due to its quick degradation by RNases in cells, while the endogenously expressed ribozyme RzC showed significant inhibitory effect on HBV replication. In conclusion, the results suggested the possibility of the hammerhead ribozyme RzC to be used for the gene therapy of HBV infection.

Hammerhead Ribozymes Suppress HBV(adr) in HepG2 Cells.

Three hammerhead ribozymes (RS3, RC2 and RC1) targeting to the HBV genome have been designed. Plasmids were constructed by inserting the genes of naked and tRNA-embedded ribozymes into RNA trimming vector pRG523 and then were transferred to eukaryotic expression vector. By the similar cloning method the shotgun-type plasmids carrying homogeneous RS3 or RtS3 unitconnected in tandem were obtained. After co-transfecting the above plasmids and HBV genome containing plasmid into human hepatoma cell line HepG2 respectively and selection by G418, the HBV-inhibiting activity of different kinds of ribozyme in G418-resistant cells was achieved by measuring the decrease of HBV-RNA, progeny DNA and the antigens expressed. The results showed that all the ribozymes were active with more than 70% inhibition activity against the HBV and that tRNA-embedded ribozymes had higher activity than naked ribozymes. It is worth particular interest that shotgun-type ribozymes with the connected unit in tandem with 8 and 12 units constructed in the plasmid revealed the highest activity, reaching >90% inhibition.

Transgenic Silkworm of Anti-NPV Ribozyme.

The plasmid pGL2Rz including ribozyme gene was linearized and introduced into early eggs of silkworm (G(0)) by gene gun. The luciferase activity in blood of the G(1) generation was detected, then the resistant silkworm was selected by NPV infection from G(2) generation. The transgenic silkworm resistant against NPV 10 times more than control ones was got at the G(4) generation. PCR and Southern blotting proved that the ribozyme gene was integrated into the genome of silkworm multicopily. The expression of ribozyme was also detected by RT-PCR in pupa. The results showed that the transgenic silkworm strain of anti-NPV ribozyme has been got.

tRNA-embedded Hammerhead Ribozymes Mediate Destruction of HBV (Subtype adr) in vitro.

Two hammerhead ribozymes, RS3 and RC2, targeting the HBsAg and HBcAg regions of HBV (adr) RNA respectively, have been designed. To improve the stability of ribozymes, two tRNA-embedded ribozymes, RtS3 and RtC2, were obtained by inserting RS3 and RC2 into the anticodon region of tRNA. Although the catalytic activity of the tRNA-embedded ribozymes was lower than that of their naked counterparts, the stability of the former in fetal bovine serum or cytoplasmic extract of HepG2 cells was much higher than that of the naked ribozymes. These results suggest that the antivirus effect of tRNA-embedded ribozymes may be greater in vivo in comparison with naked ribozymes.

Expression of Anti-PCNA mRNA Ribozyme by T7 Vaccinia Virus System in HeLa Cells.

We have designed and synthesized a hammer-head Rz targeted to human proliferating cell nuclear antigen (PCNA) mRNA at site No.524. Results showed that Rz524 can cleave the transcribed substrate RNA site-specifically in vitro. A self trimming expression plasmid of this Rz was constructed and co-introduced into HeLa cells with T7 vaccinia virus. Dot-blotting hybridization showed that the Rz was expressed in HeLa cells.

Purification and Characterization of DNA Helicase BstH1 from Bacillus stearothermophilus.

We have partially purified a DNA helicase BstH1 from Bacillus stearothermophilus through Polymin P precipitation, ammonia sulfate precipitation and column chromatographic steps with Pheny1-Sepharose, DEAE-cellulose, phosphocellulose, FPLC Mono Q and Superose 12. Bsth1 possesses a DNA-Dependent ATPase activity in the presence of Mg(2+). The ATPase activity of BstH1 is differentially stimulated by the presence of different types of nucleic acids. BstH1 has an optimal ATPase activity at 55 degrees. The DNA helicase activity of BstH1 requires a 3'-terminal single-stranded DNA binding site to initiate the unwinding reaction in the 3' right curved arrow 5'direction. BstH1 can unwind blunt-ended duplex DNA in a concentration-dependent manner.

Computer Analysis of the Cleavage Reaction of Hammerhead Ribozyme.

We have analyzed various in vitro cleavage reactions of hammerhead ribozyme by a computer program. Three energy changes: deltaE(s'), deltaE(r'), and deltaE, were calculated. deltaE(s') was the hypothetical energy requirement for the substrate to be activated from the most stable state to the relaxed state. deltaE(r') was the same energy requirement for the ribozyme. deltaE was the calculated free energy decrease by the formation of Rz-S complex from the relaxed ribozyme and the substrate. We found that high deltaE(s') or deltaE(r') reduced the cleavage efficiency of the reaction, and that the deltaE was related to both cleavage efficiency and the optimal temperature of the reaction. deltaE(s'), deltaE(r') can be calculated in advance according to the sequences of the ribozyme and the substrate. Our model, therefore, can be use in the designing of ribozymes.