Oxidative modification of neurofilament-L by the Cu,Zn-superoxide dismutase and hydrogen peroxide system

Neurofilament-L (NF-L) is a major element of neuronal cytoskeletons and known to be important for their survival in vivo. Since oxidative stress might play a critical role in the pathogenesis of neurodegenerative diseases, we investigated the role of Cu,Zn-superoxide dismutase (SOD) in the modification of NF-L. When disassembled NF-L was incubated with Cu,Zn-SOD and H2O2, the aggregation of protein was proportional to the concentration of hydrogen peroxide. Cu,Zn-SOD/H2O2-mediated modification of NF-L was significantly inhibited by radical scavenger, spin trap agents and copper chelators. Dityrosine crosslink formation was obtained in Cu,Zn-SOD/H2O2-mediated NF-L aggregates. Antioxidant molecules, carnosine and anserine significantly inhibited the aggregation of NF-L and the formation of dityrosine. This study suggests that copper-mediated NF-L modification may be closely related to oxidative reactions which play a critical role in neurodegenerative diseases.     

Acrolein, the toxic endogenous aldehyde, induces neurofilament-L aggregation

Acrolein is a highly reactive by product of lipid peroxidation and individuals with neurodegenerative disorders have been shown to contain elevated concentrations of this molecule in the brain. In the present study, we examined the pattern of neurofilament-L (NF-L) modification elicited by acrolein. When NF-L was incubated with acrolein, protein aggregation occurred in a acrolein concentration-dependent manner. Exposure of NF-L to acrolein also led to the generation of protein carbonyl compounds. Through the addition of free radical scavengers we observed a significant decrease in acrolein-mediated NF-L aggregation. These results indicate that free radicals may be involved in the modification of NF-L by acrolein. In addition, dityrosine crosslink formation was observed in acrolein-mediated NF-L aggregates and these aggregates displayed thioflavin T reactivity, reminiscent of amyloid. This study suggests that acrolein-mediated NF-L aggregation might be closely related to oxidative reactions, thus these reactions may play a critical role in neurodegenerative diseases.     

Hydrogen peroxide can be generated by tau in the presence of Cu(II)

Alzheimer's disease has been closely related with oxidative stress, which might be responsible for the dysfunction or death of neuronal cells that contributes to disease pathogenesis. Impaired copper homeostasis makes contribution to the oxidative stress and consequently to several neurodegenerative conditions. Inappropriate binding of Cu(II) to cellular proteins are currently being explored as sources of pathological oxidative stress in several neurodegenerative disorders. Here we report that a fragment of tau protein possesses copper reduction activity and initiates the copper-mediated generation of hydrogen peroxide. The tau peptide was found to be oxidized to form disulfide bond-linked dimer. The hydrogen peroxide generated was quantified by TCEP/DTNB (tris(2-carboxyethyl) phosphine hydrochloride/5,5'-dithio-bis(2-nitrobenzoic acid). Since the copper reduction capacity and the generation of hydrogen peroxide were believe to be a major toxicological pathway of Abeta peptide, the functional similarity shared by tau and Abeta implies a new perspective of tau pathology.     

Metal-catalyzed oxidation of alpha-synuclein in the presence of Copper(II) and hydrogen peroxide

alpha-Synuclein is a component of abnormal protein depositions of Lewy bodies and senile plaques found in Parkinson's and Alzheimer's diseases, respectively. By using chemical coupling reagents such as dicyclohexylcarbodiimide or N-(ethoxycarbonyl)-2-ethoxy-1, 2-dihydroquinoline, the protein was shown to experience self-oligomerization in the presence of either copper(II) or Abeta25-35. The oligomers which appeared as a ladder on a 10-20% Tricine/SDS-PAGE have been suggested to participate in the formation of protein aggregations by possibly providing a nucleation center. Since oxidatively modified protein could increase its own tendency toward protein aggregation, metal-catalyzed oxidation of alpha-synuclein has been examined with copper(II) and hydrogen peroxide in the absence of the coupling reagent. Intriguingly, the protein was also self-oligomerized into an SDS-resistant ladder on the gel. This biochemically specific copper-mediated oxidative oligomerization was shown to be dependent upon the acidic C-terminus of alpha-synuclein because the C-terminally truncated proteins such as alpha-syn114 and alpha-syn97 were not affected by the metal and hydrogen peroxide. More importantly, the oxidative oligomerization was synergistically enhanced by the presence of Abeta25-35, indicating that the peptide interaction with alpha-synuclein facilitated the copper(II) binding to the acidic C-terminus and subsequent oxidative crosslinking. It has been, therefore, suggested that abnormalities in copper and H(2)O(2) homeostasis and certain pathological factors functionally similar to the Abeta25-35 could play critical roles in the metal-catalyzed oxidative oligomerization of alpha-synuclein, which may lead to possible protein aggregation and neurodegenerations.     

Protective effects of histidine dipeptides on the modification of neurofilament-L by the cytochrome c/hydrogen peroxide system

Neurofilament-L (NF-L) is a major element of the neuronal cytoskeleton and is essential for neuronal survival. Moreover, abnormalities in NF-L result in neurodegenerative disorders. Carnosine and the related endogeneous histidine dipeptides prevent protein modifications such as oxidation and glycation. In the present study, we investigated whether histidine dipeptides, carnosine, homocarnosine, or anserine protect NF-L against oxidative modification during reaction between cytochrome c and H(2)O(2). Carnosine, homocarnosine and anserine all prevented cytochrome c/H(2)O(2)-mediated NF-L aggregation. In addition, these compounds also effectively inhibited the formation of dityrosine, and this inhibition was found to be associated with the reduced formations of oxidatively modified proteins. Our results suggest that carnosine and histidine dipeptides have antioxidant effects on brain proteins under pathophysiological conditions leading to degenerative damage, such as, those caused by neurodegenerative disorders.     

Role of cytochrome c as a stimulator of alpha-synuclein aggregation in Lewy body disease.

alpha-Synuclein is a major component of aggregates forming amyloid-like fibrils in diseases with Lewy bodies and other neurodegenerative disorders, yet the mechanism by which alpha-synuclein is intracellularly aggregated during neurodegeneration is poorly understood. Recent studies suggest that oxidative stress reactions might contribute to abnormal aggregation of this molecule. In this context, the main objective of the present study was to determine the potential role of the heme protein cytochrome c in alpha-synuclein aggregation. When recombinant alpha-synuclein was coincubated with cytochrome c/hydrogen peroxide, alpha-synuclein was concomitantly induced to be aggregated. This process was blocked by antioxidant agents such as N-acetyl-L-cysteine. Hemin/hydrogen peroxide similarly induced aggregation of alpha-synuclein, and both cytochrome c/hydrogen peroxide- and hemin/hydrogen peroxide-induced aggregation of alpha-synuclein was partially inhibited by treatment with iron chelator deferoxisamine. This indicates that iron-catalyzed oxidative reaction mediated by cytochrome c/hydrogen peroxide might be critically involved in promoting alpha-synuclein aggregation. Furthermore, double labeling studies for cytochrome c/alpha-synuclein showed that they were colocalized in Lewy bodies of patients with Parkinson's disease. Taken together, these results suggest that cytochrome c, a well known electron transfer, and mediator of apoptotic cell death may be involved in the oxidative stress-induced aggregation of alpha-synuclein in Parkinson's disease and related disorders.     

Salsolinol, a tetrahydroisoquinoline-derived neurotoxin, induces oxidative modification of neurofilament-L protection by histidyl dipeptides

Salsolinol (1-methyl-6,7-dihydroxy-1,2,3,4-tetrahydroisoquinoline) is a compound derived from dopamine metabolism and is capable of causing dopaminergic neurodegeneration. Oxidative modification of neurofilament proteins has been implicated in the pathogenesis of neurodegenerative disorders. In this study, oxidative modification of neurofilament-L (NF-L) by salsolinol and the inhibitory effects of histidyl dipeptides on NF-L modification were investigated. When NF-L was incubated with 0.5 mM salsolinol, the aggregation of protein was increased in a time-dependent manner. We also found that the generation of hydroxyl radicals (?OH) was linear with respect to the concentrations of salsolinol as a function of incubation time. NF-L exposure to salsolinol produced losses of glutamate, lysine and proline residues. These results suggest that the aggregation of NF-L by salsolinol may be due to oxidative damage resulting from free radicals. Carnosine, histidyl dipeptide, is involved in many cellular defense processes, including free radical detoxification. Carnosine, and anserine were shown to significantly prevent salsolinol- mediated NF-L aggregation. Both compounds also inhibited the generation of ?OH induced by salsolinol. The results indicated that carnosine and related compounds may prevent salsolinol-mediated NF-L modification via free radical scavenging.     

Copper-dependent oxidative stress and neurodegeneration

Copper is an essential trace element, but its redox reactivity leads to risks of damage to cell and tissues. These are well exemplified by several forms of neurodegenerative diseases, either arising as inherited disorders of copper metabolism, such as Menkes' and Wilson's disease, or as conformational diseases such as Alzheimer's disease and prion diseases. This review will cover some aspects of the involvement of copper-mediated oxidative stress in degenerative processes in the central nervous system, with special focus on the familial form of amyotrophic lateral sclerosis (FALS). Furthermore, a possible role of copper reactivity in inducing critical steps in the apoptotic pathways leading to neurodegeneration is envisaged.     

DJ-1 Is a Copper Chaperone Acting on SOD1 Activation

Lack of oxidative stress control is a common and often prime feature observed in many neurodegenerative diseases. Both DJ-1 and SOD1, proteins involved in familial Parkinson disease and amyotrophic lateral sclerosis, respectively, play a protective role against oxidative stress. Impaired activity and modified expression of both proteins have been observed in different neurodegenerative diseases. A potential cooperative action of DJ-1 and SOD1 in the same oxidative stress response pathway may be suggested based on a copper-mediated interaction between the two proteins reported here. To investigate the mechanisms underlying the antioxidative function of DJ-1 in relation to SOD1 activity, we investigated the ability of DJ-1 to bind copper ions. We structurally characterized a novel copper binding site involving Cys-106, and we investigated, using different techniques, the kinetics of DJ-1 binding to copper ions. The copper transfer between the two proteins was also examined using both fluorescence spectroscopy and specific biochemical assays for SOD1 activity. The structural and functional analysis of the novel DJ-1 copper binding site led us to identify a putative role for DJ-1 as a copper chaperone. Alteration of the coordination geometry of the copper ion in DJ-1 may be correlated to the physiological role of the protein, to a potential failure in metal transfer to SOD1, and to successive implications in neurodegenerative etiopathogenesis.     

Mitochondrial superoxide dismutase SOD2, but not cytosolic SOD1, plays a critical role in protection against glutamate-induced oxidative stress and cell death in HT22 neuronal cells

Oxidative cell death is an important contributing factor in neurodegenerative diseases. Using HT22 mouse hippocampal neuronal cells as a model, we sought to demonstrate that mitochondria are crucial early targets of glutamate-induced oxidative cell death. We show that when HT22 cells were transfected with shRNA for knockdown of the mitochondrial superoxide dismutase (SOD2), these cells became more susceptible to glutamate-induced oxidative cell death. The increased susceptibility was accompanied by increased accumulation of mitochondrial superoxide and loss of normal mitochondrial morphology and function at early time points after glutamate exposure. However, overexpression of SOD2 in these cells reduced the mitochondrial superoxide level, protected mitochondrial morphology and functions, and provided resistance against glutamate-induced oxidative cytotoxicity. The change in the sensitivity of these SOD2-altered HT22 cells was neurotoxicant-specific, because the cytotoxicity of hydrogen peroxide was not altered in these cells. In addition, selective knockdown of the cytosolic SOD1 in cultured HT22 cells did not appreciably alter their susceptibility to either glutamate or hydrogen peroxide. These findings show that the mitochondrial SOD2 plays a critical role in protecting neuronal cells from glutamate-induced oxidative stress and cytotoxicity. These data also indicate that mitochondria are important early targets of glutamate-induced oxidative neurotoxicity.     

Characterizing effects of excess copper levels in a human astrocytic cell line with focus on oxidative stress markers

BackgroundBeing an essential trace element, copper is involved in diverse physiological processes. However, excess levels might lead to adverse effects. Disrupted copper homeostasis, particularly in the brain, has been associated with human diseases including the neurodegenerative disorders Wilson and Alzheimer’s disease. In this context, astrocytes play an important role in the regulation of the copper homeostasis in the brain and likely in the prevention against neuronal toxicity, consequently pointing them out as a potential target for the neurotoxicity of copper. Major toxic mechanisms are discussed to be directed against mitochondria probably via oxidative stress. However, the toxic potential and mode of action of copper in astrocytes is poorly understood, so far.MethodsIn this study, excess copper levels affecting human astrocytic cell model and their involvement in the neurotoxic mode of action of copper, as well as, effects on the homeostasis of other trace elements (Mn, Fe, Ca and Mg) were investigated.ResultsCopper induced substantial cytotoxic effects in the human astrocytic cell line following 48 h incubation (EC₃₀: 250 μM) and affected mitochondrial function, as observed via reduction of mitochondrial membrane potential and increased ROS production, likely originating from mitochondria. Moreover, cellular GSH metabolism was altered as well. Interestingly, not only cellular copper levels were affected, but also the homeostasis of other elements (Ca, Fe and Mn) were disrupted.ConclusionOne potential toxic mode of action of copper seems to be effects on the mitochondria along with induction of oxidative stress in the human astrocytic cell model. Moreover, excess copper levels seem to interact with the homeostasis of other essential elements such as Ca, Fe and Mn. Disrupted element homeostasis might also contribute to the induction of oxidative stress, likely involved in the onset and progression of neurodegenerative disorders. These insights in the toxic mechanisms will help to develop ideas and approaches for therapeutic strategies against copper-mediated diseases.     

Post-aggregation Oxidation of Mutant Huntingtin Controls the Interactions between Aggregates

Aggregation of protein molecules is a pathological hallmark of many neurodegenerative diseases. Abnormal modifications have often been observed in the aggregated proteins, supporting the aggregation mechanism regulated by post-translational modifications on proteins. Modifications are in general assumed to occur in soluble proteins before aggregation, but actually it remains quite obscure when proteins are modified in the course of the aggregation. Here we focus upon aggregation of huntingtin (HTT), which causes a neurodegenerative disorder, Huntington disease, and we show that oxidation of a methionine residue in HTT occurs in vitro and also in vivo. Copper ions as well as added hydrogen peroxide are found to oxidize the methionine residue, but notably, this oxidative modification occurs only in the aggregated HTT but not in the soluble state. Furthermore, the methionine oxidation creates additional interactions among HTT aggregates and alters overall morphologies of the aggregates. We thus reveal that protein aggregates can be a target of oxidative modifications and propose that such a “post-aggregation” modification is a relevant factor to regulate properties of protein aggregates.     

alpha-Synuclein protects against oxidative stress via inactivation of the c-Jun N-terminal kinase stress-signaling pathway in neuronal cells.

The expression of alpha-synuclein, a synaptic molecule implicated in the pathogenesis of neurodegenerative disorders such as Parkinson's disease and Lewy body disease is increased upon injury to the nervous system, indicating that it might play a role in regeneration and plasticity; however, the mechanisms are unclear. Because c-Jun N-terminal kinase (JNK), a member of the mitogen-activated protein kinase family, plays an important role in stress response, the main objective of the present study was to better understand the involvement of this pathway in the signaling responses associated with resistance to injury in cells expressing alpha-synuclein. For this purpose, the JNK-signaling pathway was investigated in alpha-synuclein-transfected neuronal cell line glucose transporter (GT) 1-7 under oxidative stress conditions. Although hydrogen peroxide challenge resulted in JNK activation and cell death in cells transfected with vector control or beta-synuclein, alpha-synuclein-transfected cells were resistant to hydrogen peroxide, and JNK was not activated. The inactivation of JNK in the alpha-synuclein-transfected cells was associated with increased expression and activity of JNK-interacting protein (JIP)-1b/islet-brain (IB)1, the scaffold protein for the JNK pathway. Similarly, cells transfected with JIP-1b/IB1 were resistant to hydrogen peroxide associated with inactivation of the JNK pathway. In these cells, expression of endogenous alpha-synuclein was significantly increased at the protein level. Furthermore, alpha-synuclein was co-localized with JIP-1b/IB1 in the growth cones. Taken together, these results suggest that increased alpha-synuclein expression might protect cells from oxidative stress by inactivation of JNK via increased expression of JIP-1b/IB1. Furthermore, interactions between alpha-synuclein and JIP-1b/IB1 may play a mutual role in the neuronal response to injury and neurodegeneration.     

Cupric Ions Induce the Oxidation and Trigger the Aggregation of Human Superoxide Dismutase 1

Background

Amyotrophic lateral sclerosis (ALS), partly caused by the mutations and aggregation of human copper, zinc superoxide dismutase (SOD1), is a fatal degenerative disease of motor neurons. Because SOD1 is a major copper-binding protein present at relatively high concentration in motor neurons and copper can be a harmful pro-oxidant, we want to know whether aberrant copper biochemistry could underlie ALS pathogenesis. In this study, we have investigated and compared the effects of cupric ions on the aggregation of ALS-associated SOD1 mutant A4V and oxidized wild-type SOD1.

Methodology/Principal Findings

As revealed by 90° light scattering, dynamic light scattering, SDS-PAGE, and atomic force microscopy, free cupric ions in solution not only induce the oxidation of either apo A4V or Zn₂-A4V and trigger the oligomerization and aggregation of oxidized A4V under copper-mediated oxidative conditions, but also trigger the aggregation of non-oxidized form of such a pathogenic mutant. As evidenced by mass spectrometry and SDS-PAGE, Cys-111 is a primary target for oxidative modification of pathological human SOD1 mutant A4V by either excess Cu²⁺ or hydrogen peroxide. The results from isothermal titration calorimetry show that A4V possesses two sets of independent binding sites for Cu²⁺: a moderate-affinity site (10⁶ M^-1) and a high-affinity site (10⁸ M^-1). Furthermore, Cu²⁺ binds to wild-type SOD1 oxidized by hydrogen peroxide in a way similar to A4V, triggering the aggregation of such an oxidized form.

Conclusions/Significance

We demonstrate that excess cupric ions induce the oxidation and trigger the aggregation of A4V SOD1, and suggest that Cu²⁺ plays a key role in the mechanism of aggregation of both A4V and oxidized wild-type SOD1. A plausible model for how pathological SOD1 mutants aggregate in ALS-affected motor neurons with the disruption of copper homeostasis has been provided.     

Hydrogen peroxide attenuates insulin-like growth factor-1 neuroprotective effect, prevented by minocycline

Oxidative stress-induced neuronal death due to hydrogen peroxide overload plays a critical role in the pathogenesis of numerous neurological diseases. Insulin-like growth factor-1 (IGF-1) is important in maintaining neuronal survival, proliferation, and differentiation in the central nervous system. We now report that sublethal doses of hydrogen peroxide attenuated IGF-1 neuroprotective activity on cultured cerebellar granule neurons under potassium and serum deprivation. Interestingly, this attenuation can be prevented by minocycline, an antibiotic that has been shown to have neuroprotective activity in animal models of neuronal injury. Furthermore, hydrogen peroxide also blocked IGF-1's neuroprotection for cortical neurons deprived of neurotrophic factors (B27), which was prevented by minocycline. Our data suggest that inhibition of IGF-1 signaling by hydrogen peroxide may constitute an additional pathway contributing to its neurotoxicity. More importantly, combining minocycline and IGF-1 could be an effective treatment in neurological diseases associated with both oxidative stress and deficiency of IGF-1.     

Mitochondrial Dysfunction in Neurodegenerative Diseases Associated with Copper Imbalance

Copper is an essential transition metal ion for the function of key metabolic enzymes, but its uncontrolled redox reactivity is source of reactive oxygen species. Therefore a network of transporters strictly controls the trafficking of copper in living systems. Deficit, excess, or aberrant coordination of copper are conditions that may be detrimental, especially for neuronal cells, which are particularly sensitive to oxidative stress. Indeed, the genetic disturbances of copper homeostasis, Menkes' and Wilson's diseases, are associated with neurodegeneration. Furthermore, copper interacts with the proteins that are the hallmarks of neurodegenerative disorders, such as Alzheimer's disease, Parkinson's disease, prion diseases, and familial amyotrophic lateral sclerosis. In all cases, copper-mediated oxidative stress is linked to mitochondrial dysfunction, which is a common feature of neurodegeneration. In particular we recently demonstrated that in copper deficiency, mitochondrial function is impaired due to decreased activity of cytochrome c oxidase, leading to production of reactive oxygen species, which in turn triggers mitochondria-mediated apoptotic neurodegeneration.     

Hydrogen peroxide is generated during the very early stages of aggregation of the amyloid peptides implicated in Alzheimer disease and familial British dementia

Alzheimer disease and familial British dementia are neurodegenerative diseases that are characterized by the presence of numerous amyloid plaques in the brain. These lesions contain fibrillar deposits of the beta-amyloid peptide (Abeta) and the British dementia peptide (ABri), respectively. Both peptides are toxic to cells in culture, and there is increasing evidence that early "soluble oligomers" are the toxic entity rather than mature amyloid fibrils. The molecular mechanisms responsible for this toxicity are not clear, but in the case of Abeta, one prominent hypothesis is that the peptide can induce oxidative damage via the formation of hydrogen peroxide. We have developed a reliable method, employing electron spin resonance spectroscopy in conjunction with the spin-trapping technique, to detect any hydrogen peroxide generated during the incubation of Abeta and other amyloidogenic peptides. Here, we monitored levels of hydrogen peroxide accumulation during different stages of aggregation of Abeta-(1-40) and ABri and found that in both cases it was generated as a short "burst" early on in the aggregation process. Ultrastructural studies with both peptides revealed that structures resembling "soluble oligomers" or "protofibrils" were present during this early phase of hydrogen peroxide formation. Mature amyloid fibrils derived from Abeta-(1-40) did not generate hydrogen peroxide. We conclude that hydrogen peroxide formation during the early stages of protein aggregation may be a common mechanism of cell death in these (and possibly other) neurodegenerative diseases.     

Cell-permeable peptide antioxidants as a novel therapeutic approach in a mouse model of amyotrophic lateral sclerosis

Reactive oxygen species (ROS) play a major role in the pathogenesis of neurodegenerative diseases. They are important contributors to necrotic and apoptotic cell death. A major proportion of cellular ROS is generated at the inner mitochondrial membrane by the respiratory chain. In the present study, we investigated a novel peptide antioxidant (SS-31) targeted to the inner mitochondrial membrane for its therapeutic effects both in vitro and in vivo in the G93A mouse model of amyotrophic lateral sclerosis (ALS). SS-31 protected against cell death induced by hydrogen peroxide in vitro in neuronal cells stably transfected with either wild-type or mutant Cu/Zn superoxide dismutase (SOD1). Daily intraperitoneal injections of SS-31 (5 mg/kg), starting at 30 days of age, led to a significant improvement in survival and motor performance. In comparison with vehicle-treated G93A mice, SS-31-treated mice showed a decreased cell loss and a decrease in immunostaining for markers of oxidative stress in the lumbar spinal cord. This further enhances the concept that pharmacological modification of oxidative stress is a therapeutic option for the treatment of ALS.     

Oxidative modification of M-type K(+) channels as a mechanism of cytoprotective neuronal silencing

Voltage-gated K(+) channels of the Kv7 family underlie the neuronal M current that regulates action potential firing. Suppression of M current increases excitability and its enhancement can silence neurons. We here show that three of five Kv7 channels undergo strong enhancement of their activity by oxidative modification induced by physiological concentrations of hydrogen peroxide. A triple cysteine pocket in the channel S2-S3 linker is critical for this effect. Oxidation-induced enhancement of M current produced a hyperpolarization and a dramatic reduction of action potential firing frequency in rat sympathetic neurons. As hydrogen peroxide is robustly produced during hypoxia-induced oxidative stress, we used an oxygen/glucose deprivation neurodegeneration model that showed neuronal death to be severely accelerated by M current blockade. Such blockade had no effect on survival of normoxic neurons. This work describes a novel pathway of M-channel regulation and suggests a role for M channels in protective neuronal silencing during oxidative stress.     

Contribution of a metal-peroxide adduct to neurodegeneration is due to its oxidative protease activity

Many hypotheses have been developed to explain aging and age-related neurodegenerative disorders; one of the most compelling is the role of oxidative stress to induce changes in protease activity in brains of patients of Alzheimer's disease and prion disease. At the moment however, there is no clear answer how protein degradation may be achieved in the brain. We have observed that several metal compounds can degrade proteins in the presence of hydrogen peroxide, and elucidated the reaction scheme based on the new theoretical point for the reactivity of a metal-peroxide adduct with eta 1-coordination mode. In this article we would like to point out the importance of a copper(II)-peroxide adduct to promote neurodegenerative diseases such as prion disease and amyotrophic lateral sclerosis through its oxidative protease function.