In this thesis, I have elucidated the individual roles of four HSFA1 member and HSFA2.
The multiplicity and redundancy of plant HSFs indeed complicate their functional genetic studies. The phenotypic effects of single KO mutation of HSFs are weak due to the functional redundancy. For the case of HSFA1s in Arabidopsis, the thermotolerance phenotypes were not obvious whether in single or double KO mutants. It may be due to the functional redundancy or weak contribution in thermotolerance of the HSFA1 genes. Therefore, generating the triple and even quadruple mutants of HSFA1s is required to address their overall and individual functions.
In Chapters 2 and 3, I describe the generation of the triple and quadruple mutants of HSFA1s, and partition of HSR as HSFA1s-dependent and HSFA1s-independent parts. We also show the novel function of HSFA1s in the development of seeds and seedlings.
For the individual roles of HSFA2, the study on single KO mutant already demonstrated its unique function in extending the acquired thermotolerance. However, it is not clearly addressed whether HSFA2 could trigger HSR without HSFA1s. In chapter 4, I describe the effects of constitutively expressed HSFA2 in the quadruple mutant of HSFA1s, and
demonstrate the novel role of HSFs in the chronic heat stress at temperature as low as 27°C.
Chapter 2
The role of class A1 heat shock factors (HSFA1s) in response to heat and other stresses in Arabidopsis.
The contents of this chapter are published in Plant, Cell & Environment (Liu et al., 2011).
2.1 Abstract
In Arabidopsis, there are four homologs of class A1 heat shock factor (HSFA1) genes, which likely encode the master regulators of heat shock response (HSR). However, previous studies with double knockout (KO) mutants were unable to confirm this point probably due to functional redundancy. Here, we generated a quadruple KO (QK) and four triple KO mutants to dissect their functions. Our data show that members of the HSFA1 group not only play a
pivotal role in HSR but also are involved in growth and development. Alterations in morphology and retardation in growth were observed in the quadruple but not in triple KO mutants. The basal and acquired thermotolerance capacity was dramatically decreased in the QK mutant but varied in triple KO mutants at different developmental stages. The
transcriptomics profiles suggested that more than 65% of the heat stress (HS)-up-regulated genes were HSFA1 dependent. HSFA1s were also involved in the expression of several HS genes induced by H2O2, salt and mannitol, which is consistent with the increased sensitive phenotype of the QK mutant to the stress factors. In conclusion, the Arabidopsis HSFA1s function as the master regulators of HSR and participate as important components in other abiotic stress responses as well.
2.2 Introduction
In eukaryotes, the conserved heat shock transcription factors (HSFs) are the major regulators of heat stress (HS) responsive genes encoding molecular chaperones and other stress proteins (Wu, 1995). Higher plant genomes contain an expanded HSF family as compared to other eukaryotes (Nover et al., 2001; Shiu et al., 2005). For example, Arabidopsis and rice have at least 21 and 25 HSF homologues, respectively (Nover et al., 2001; Wang et al., 2009).
However, to date, only one and four HSFs can be found in the genomes of invertebrates and vertebrates, respectively (Åkerfelt et al., 2010). The large HSF family in higher plants may be associated with fitness of the sessile organisms that have to constantly face recurring or multiple environmental challenges including extreme temperature, light intensity, flooding, drought, and soil salinity. Understanding the function and complexity of HSF network in higher plants is important in the contexts of the global climate change, which may significantly increase the frequency of duration of extreme heat in some regions (Hayhoe et al., 2004).
Based on structural characteristics and phylogenetic analysis, Arabidopsis HSFs are allocated into three major classes (class A, B, and C) and 14 groups (A1-9, B1-4, and C1;
(Nover et al., 2001). The rice HSF family has representatives in most of the groups (von Koskull-Döring et al., 2007; Wang et al., 2009). Moreover, the number of HSFs in each group varies in different species. For example, rice possesses only one member in the group A1, while Arabidopsis has four (A1a, b, d, e) and tomato has at least three members (A1a, b, c). By contrast, Arabidopsis has only one member in the group A2, whereas rice contains five
members (A2a-e) (von Koskull-Döring et al., 2007). In general, class A HSFs but not class B or class C HSFs contain a transactivator motif (Nover et al., 2001; Kotak et al., 2004).
Due to the availability of T-DNA knockout (KO) lines, considerable progress has been
knockdown of HSFA1a expression dramatically reduce thermotolerance in tomato (Mishra et al., 2002). In Arabidopsis, HSFA1a and HSFA1b were shown to work redundantly on a subset of HS response genes in the early phase of HS response (Lohmann et al., 2004; Busch et al., 2005). However, the HSFA1a/1b double KO mutant does not show substantial defects in thermotolerance (Lohmann et al., 2004). Thus, a unanimous agreement on whether HSFA1 act as the master regulator of HSR in plants has not yet been reached.
Among the HSF family, HSFA2 is the most heat-inducible one (Busch et al., 2005). Its expression is suppressed in the HSFA1a knockdown tomato plants (Mishra et al., 2002).
However, the expression of Arabidopsis HSFA2 is not affected in the HSFA1a/1b double KO line (Busch et al., 2005). Recent studies pointed out the pivotal role of HSFA2 in the late phase of HSR (Schramm et al., 2006; Charng et al., 2007; Wunderlich et al., 2007), suggesting that HSFA2 is a secondary regulator under the control of at least one master regulator to form a transcriptional cascade or network that covers the early and late expression of HS genes.
HSFA2 was also shown to be involved in other environmental stresses (Nishizawa et al., 2006;
Banti et al., 2010). Recently, components that modulate the activities of HSFA1 and A2 in Arabidopsis were reported. Calmodulin-binding protein kinase 3 (AtCBK3) was shown to phosphorylate HSFA1a in vitro and enhance the binding activity of HSF to the heat shock element (Liu et al., 2008). The HSF binding protein (HSBP) was shown to interact with HSFA1a, HSFA1b and HSFA2 and negatively regulate HSR (Hsu et al., 2010). ROF1 and ROF2, which are homologs of peptidyl prolyl cis/trans isomerase, were shown to play positive and negative roles, respectively, in regulating the activity of HSFA2 (Meiri and Breiman, 2009;
Meiri et al., 2010).
Arabidopsis HSFA3 is a HS-induced gene, which is under the control of DREB2A, a HS-inducible gene itself that functions in both HS and water-deficit stress response (Sakuma et al., 2006; Schramm et al., 2008). A moderate defect in thermotolerance was observed in the
DREB2A KO as well as the HSFA3 KO mutants (Sakuma et al., 2006; Schramm et al., 2008;
Yoshida et al., 2008). So far, the upstream regulator of DREB2A and HSFA3 under HS condition remains to be identified.
Biological functions of the members of the A4 group have been demonstrated in rice (Shim et al., 2009). The spl7 mutant of OsHSFA4d leads to spotted leaf phenotype under elevated temperature and solar radiation (Yamanouchi et al., 2002). Recent studies show that OsHSFA4a is involved in cadmium tolerance (Shim et al., 2009). In Arabidopsis,
overexpressing a dominant-negative HSFA4a suppresses the expression of the cytosolic H2O2-scavenging ASCORBATE PEROXIDASE 1 (APX1) that is induced under moderate light stress (Davletova et al., 2005). It was postulated that HSFA4a acts as an H2O2 sensor (Miller and Mittler, 2006).
Arabidopsis HSFA9 is regulated by the seed-specific transcription factor ABI3 and is expressed exclusively in the late stages of seed development (Kotak et al., 2007). It has been suggested that HSFA9 plays a specialized role in regulating HSP genes during seed maturation (Kotak et al., 2007). In agree with this idea, overexpression of sunflower HSFA9 fused a transcription repressor motif in transgenic tobacco caused a reduction in the levels of seed HSPs and seed basal thermotolerance (BT) (Tejedor-Cano et al., 2010).
Compared to the class A HSFs, the biological functions of class B and class C HSFs are less understood. Tomato HSFB1 was shown to be a general transcription co-activator that interacts with HAC1, a plant ortholog of CREB-binding protein (Bharti et al., 2004). However, there are evidences showing that members of the class B HSFs act as transcriptional repressors in Arabidopsis (Czarnecka-Verner et al., 2000; Ikeda and Ohme-Takagi, 2009; Kumar et al., 2009) and soybeans (Czarnecka-verner et al., 2004). Recently, HSFB1 and HSFB2b were
Identifying the master HSF(s) in Arabidopsis is essential in deciphering the complex crosstalks between different stress components known in this important model plant. In this study, we generated a quadruple KO (QK) and four triple KO lines, of which the latter contain only one intact HSFA1 gene. We found that HSFA1a, A1b and A1d work redundantly as the master regulators of HSR. A large portion of the HSR genes is directly or indirectly regulated by these HSFs. They were also required for the induction of several heat-induced transcription factors such as HSFA2 and DREB2A. Moreover, the induction of HSP genes by salt, osmotic, and oxidative stresses were dependent on the function of these HSFs. Interestingly, we also observed a role of these HSFs in early Arabidopsis development. To our knowledge, this is the first report on the role of HSFs on plant growth and development under well-controlled
environmental conditions.
2.3 Materials and methods
2.3.1 Plant materials and growth conditions
The HSFA1a/1b double KO and wild-type (wt) seeds in the ecotype Wassilewskija (Ws) background were kindly provided by Dr Fritz Schöffl (University of Tübingen, Germany) (Lohmann et al., 2004; Busch et al., 2005). The HSFA1d/1e double KO line was generated by crossing the T-DNA insertion line SALK_022404 (hsfA1d) and SALK_094943 (hsfA1e) obtained from the Arabidopsis Biological Resource Center (Ohio State University). The
Arabidopsis Genome Initiative (AGI) numbers of HSFA1a, A1b, A1d and A1e are AT4G17750, AT5G16820, AT1G32330 and AT3G02990, respectively. The QK and triple KO mutants were obtained by crossing the HSFA1a/1b and HSFA1d/1e double KO mutants. The T-DNA
insertions were confirmed by PCR amplification, and then homozygous lines of the mutant allele were isolated. SALK_066374 for HSP101 (AT1G74310) was obtained as previously described (Charng et al., 2006). The sequences of gene-specific primers used in this study are listed in Table 1. For propagating seeds, Arabidopsis plants in soil were grown in walk-in growth chambers at 22°C and 16 h of light (120 mol m-2 s-1). For seedlings grown in plates, Arabidopsis seeds were sterilized and sown on solid medium (25 mL of 0.5x Murashige and Skoog (MS) medium containing 0.8% agar and1%sucrose) in a Petri dish (90 x 15 mm),
incubated for 3 d at 4°C for synchronized germination, then grown to desired stage at 22°C and 16 h of light (120 mol m-2 s-1).
2.3.2 Evaluation of growth rate and seed size
The growth rate of the wt and mutants from the seed to the four rosette leaves stage was measured on agar plates containing 0.5x MS medium and 1% sucrose. Growth stages of
Table 2.1 Primers used in this study.
Target Genes Primer name Oligo Sequence Purpose
HSFA1a HSFA1a-5 AAGAAGATAAGCCGGAGAAAATCT Genotyping and RT-PCR
HSFA1a-6 ACAAAGTTGCAACCGTACTACTGA Genotyping and RT-PCR JL-202 of hsfa1a CATTTTATAATAACGCTGCGGACATCTAC Genotyping and RT-PCR
HSFA1b HSFA1b-5 CCAGCTTCGTCAGACAGTTAAATA Genotyping and RT-PCR
HSFA1b-6 TAGGAAACTGTCAGGATTGTTTGA Genotyping and RT-PCR LB of hsfa1b GATGCACTCGAAATCAGCCAATTTTAGAC Genotyping and RT-PCR
HSFA1d HSFA1d-5 GCATAATAATTTCTCCAGCTTCGT Genotyping and RT-PCR
HSFA1d-6 AGGTTTTCGCCTAGTTATTGATTG Genotyping and RT-PCR
HSFA1e HSFA1e-5 TTTTAAGAGGCCAAAAGCAAATAC Genotyping and RT-PCR
HSFA1e-6 GTTGATTCTTGCTCCACACATTAC Genotyping and RT-PCR
ABI3 ABI3-2F CCTGGATGTATTGGCCTAATGT RT-PCR
ABI3-2R AGAAACCGCAAATTCTTTTCTG RT-PCR
ACTIN2 ACT-F CGCTCTTTCTTTCCAAGCTCAT qRT-PCR and RT-PCR
ACT-R GCAAATCCAGCCTTCACCAT qRT-PCR
ATACT2-2 GTAGTCAACAGCAACAAAGGAGAGC RT-PCR
APX2 APX2-F CGTGGTCTTATTGCCGAGAAG RT-PCR
APX2-R CCCAAACGGTCCTCCTGTCT RT-PCR
DREB2A ATDREB2A-2F AAGGATTTGGGGTAAATGGGTTG qRT-PCR and RT-PCR
ATDREB2A-2R CAGCCTCATCATAAGCAGAAGCA qRT-PCR and RT-PCR
HSFA2 ATHSFA2R-1 CCATGGAAGAACTGAAAGTGGAAATGGAGG RT-PCR
ATHSFA2R-2 GCGGCCGCAGGTTCCGAACCAAG RT-PCR
HSFA2 A2-1 (AtHSFA2-F-ABI) GGTTCTGTAGCGGCTTCTTCAT qRT-PCR
A2-2 (AtHSFA2-R-ABI) TGGTGGCCCTGTTTCGTTA qRT-PCR
HSFA2 endo-1 (AtHsfA2-rlt-1) CATGGTTGATCAAATGGGTTTTC qRT-PCR
endo-2 (AtHsfA2-rlt-2) CATTACCATAACTTAGACCGCAACAA qRT-PCR
HSFA3 ATHSFA3-2F TGCTGCAGTAGCACTAGCCAAAG qRT-PCR
ATHSFA3-2R CTGCTGAAGCTCAACCATTTCCT qRT-PCR
HSFA7a ATHSFA7a-1 ATGATGAACCCGTTTCTCCCGG RT-PCR
ATHSFA7a-2 GGAGGTGGAAGCCAAACTCTCATC RT-PCR
HSFA7b ATHSFA7b-3 CTCGGCCACTATTCTGCCTCT RT-PCR
ATHSFA7b-4 TCTCCACACCATCAGTCCGTT RT-PCR
Target Genes Primer name Oligo Sequence Purpose
HSFB1 ATHSFB1-1 ATGACGGCTGTGACGGCG RT-PCR
ATHSFB1-2 GCAGACTTTGCTGCTTTTCCA RT-PCR
HSFB2a ATHSFB2a-1 ATGAATTCGCCGCCGGTTGACG RT-PCR
ATHSFB2a-2 CAACCACGGCGTCTCCTCATCGG RT-PCR
HSFB2b ATHSFB2b-1 ATGCCGGGGGAACAAACCGG RT-PCR
ATHSFB2b-2 CCGAGTTCAAGCCACGACCC RT-PCR
HSP17.6-CI ATHSP17.6-1 ATGTCTCTAATTCCAAGCATCTTCG RT-PCR
ATHSP17.6-2 TTAACCAGAGATATCAATGGACTTAAC RT-PCR
HSP17.7-CII HSP17.7-CII-F GCAAGAAAGTTAACACAA RT-PCR
HSP17.7-CII-R CACGATCACAAACAAACTC RT-PCR
HSP25.3-p HSP21-F-EJ TGGACGTCTCTCCTTTCGGAT RT-PCR
HSP21-R-EJ TGATCGAGTCCTACTGAATCTGGA RT-PCR
mtHSC70-2 At mtHsc70-2-1F CTCTATCGCTCGGTATTGAAAC qRT-PCR
At mtHsc70-2-1R CTCTTCTTTGTGGGGATGGTTG qRT-PCR
HSP90.1 ATHSP90.1-1F AATACGCTGTTGGACAATTGAA RT-PCR
ATHSP90.1-1R GATTCTCGAAGGACTTCTTCTTC RT-PCR
HSP90 HSP90-1 TTTGGTGTTGGTTTCTACTCTGCTTA qRT-PCR and RT-PCR
HSP90-2 TCGTTCTGACCTTCCTTCATCCTTGT qRT-PCR and RT-PCR
HSA32 HSA32-F-ABI GGAAGAGTTTCGAGGAGAACGA qRT-PCR
HSA32-R-ABI GACCTCGCATCTCCGTAACAC qRT-PCR
HSA32 HSA32-F-EJ AAAGACTATGTGGAGGAGTG RT-PCR
HSA32-R-EJ CACATAGAGATTCACATTTG RT-PCR
HSP101 HSP101-F-ABI TGCATTTAGCTGGTGCTTTGAT qRT-PCR
HSP101-R-ABI CCACCGGCACTAGAGATTGC qRT-PCR
HSP101 Hsp101-F-EJ CACCAGGGTATGTTGGTCACG RT-PCR
Hsp101-R EJ GCACCATACACCGGGTCATAA RT-PCR
TIL1 TIL1-F (lipocalin-F) TTTATCGAAGGCAGCGCCTATA qRT-PCR
TIL1-R (lipocalin-R) GCTTGGCTTCGTCGCTTTTA qRT-PCR
MIPS2 AT2G22240-1F GTAGCTAGTAATGGCATCCTCTTTGA qRT-PCR
were equal to or greater than 2 mm long when the primary florescence reached 1 cm in length, as described previously (Wu et al., 2008). Six individual plants for each line grown in soil pots were measured. The seed size was measured by using 50 and 60 mesh sieves (Sigma-Aldrich, St Louis, MO, USA). The exclusion sizes of the 50 and 60 mesh sieves are 297 and 250 mm, respectively. Seed were divided into three sizes: >297 mm, 297~250 mm and <250 mm.
2.3.3 Thermotolerance assays
For thermotolerance assays, seeds were sown on 0.5x MS plates with 1% sucrose and imbibed at 4°C for 3 d in the dark. After imbibition, the plates were first incubated for 2 h at 22°C and then placed for 2 h in a water bath (44°C). The germination rate was measured after recovery under normal growth conditions for 2-14 d. For the BT assay, 7-day-old seedlings were treated for 26 min at 43°C. For the short-term acquired thermotolerance (SAT) assay, 7-day-old seedlings were acclimated for 1 h at 37°C, recovered for 2 h at 22°C and then treated for 140 min at 44°C. For the long-term acquired thermotolerance (LAT) assay, 5-day-old seedlings were acclimated for 1 h at 37°C, recovered for 2 d at 22°C and then treated for 50 min at 44°C. The above heat treatments were performed in the dark to avoid photo-oxidative stress, and the details were described in (Charng et al., 2006). For the tolerance against moderately high temperature (TMHT) assay, 7-day-old seedlings were treated for 7 d at 35°C under a light/dark cycle of 16 h/8 h (120 mol m-2 s-1). After applying the different HS regimes, plants were recovered for 7 d at 22°C. At the end of recovery, pictures were taken and the survival rates were measured. For thermotolerance assays of adult plants grown in potted soil, 15-day-old plants were placed in a growth chamber and subjected to TMHT conditions. After treatment for 5 d at 35°C, the plants were moved to room temperature and pictures were taken.
2.3.4 Oxidative, osmotic and salt stress treatments
For measurement of the transcript levels of HSR genes after oxidative, osmotic or salt stress treatments, 7-day-old seedlings sown on 0.5x MS plates with 1% sucrose under normal conditions were transferred into solutions containing 150 mM NaCl, 300 mM mannitol or 5 mM H2O2 for 2 h treatment. Subsequently, samples were frozen in liquid nitrogen for RNA extraction. For germination rate assays, seeds were sown in 0.5x MS plates containing 1%
sucrose with 300 mM mannitol or 150 mM NaCl, or 0.1% sucrose with 10 mM H2O2, imbibed at 4°C in the dark for 3 d, and then grown at 22°C for 2, 4 and 7 d. For survival rate assays, seeds were sown on 0.5x MS medium containing 1% sucrose with 300 mM mannitol, 100 mM NaCl or 0.1% sucrose with 5 mM H2O2, imbibed at 4°C for 3 d, and then grown at 22°C for 15 d.
2.3.5 RNA isolation and RT-PCR analysis
Total RNA of plant tissues was isolated as previously described (Charng et al., 2006).
The RNA quantity of samples was determined on a NanoDrop ND-1000 spectrophotometer (Thermo Scientific, Waltham, MA, USA). Semi-quantitative RT-PCR analysis of tested genes was performed as previously described (Charng et al., 2007). PCR reactions were performed for 27 cycles in a volume of 10 μL using cDNA obtained from 12 ng total RNA. The
sequences of the primers for RT-PCR analysis for each gene are described in Supplemental Table 1. Quantitative RT-PCR (qRT-PCR) was performed in a 7300 Real-Time PCR System using the SYBR Green RT-PCR reagent kit following the manufacturer’s protocol (Applied Biosystems, Carlsbad, CA, USA). Each reaction was run in triplicates in a volume of 20 mL with an initial denaturation step at 95°C for 10 min, followed by 40 cycles of 95°C for 15 s and 60°C for 60 s. Data were analyzed according to the manufacturer’s instructions using the 7300
ACTIN2 from the CT value of the GOI. The amplification efficiency (E), calculated based on a previous method (Peirson et al., 2003), was larger than 0.95 in each reaction.
2.3.6 Microarray analysis
Seven-day-old seedlings of Col-0, Ws and QK mutant grown at 22°C on 0.5x MS plates containing 1% sucrose were incubated for 1 h at 37°C. Subsequently, samples were collected for RNA extraction. The experiment was repeated and two biological replicates were processed for analysis. RNA extraction, quantification and microarray analysis using the ATH1
GeneChip arrays containing 22 810 probe sets (Affymetrix, Santa Clara, CA, USA) were performed as previously described (Chi et al., 2009). Assessment of experimental quality and statistical analyses were performed using remote analysis computation for gene expression (RACE) (Psarros et al., 2005), where individual probe results files (CEL files) were used as input and normalized using the robust multi-array average (RMA) algorithm (Irizarry et al., 2003). Four thousand two hundred eighty-one features with an absent detection call and 64 control features were removed for all chips analyzed. In total, data for 18,465 probe sets were used as input data for statistical analysis. Genes whose expression changed more than threefold between mutant and wt seedlings at a P value of <0.05 were selected. False discovery rate (FDR) at the stringent level of 0.05 was used to declare statistical significance and account for multiple tests. The relative change of these 18,465 features between wt and QK mutant with the same selection criteria was compared. Annotations and gene ontology classifications of
Arabidopsis genes were obtained from The Arabidopsis Information Resource database (TAIR, http://www.Arabidopsis.org). The microarray data reported here can be accessed in the Gene Expression Omnibus at the National Center for Biotechnology Information (NCBI; accession number GSE26266).
2.3.7 Immunoblotting
Protein extraction and immunoblotting were performed as previously described (Charng et al., 2006). Rabbit antiserum against AtHSFA2 was produced by employing a synthetic peptide with the sequence ‘HLLKNIKRRRNMGLQNVN’ as antigen. Immunization and serum collection were performed by commercial service (LTK Biotechnology, Taoyuan, Taiwan). The specificity of the AtHSFA2 antibody was confirmed by probing against protein samples extracted from heat-treated HSFA2 KO plants. The antibodies against HSP101, HSP90, HSA32, sHSP-CI and tubulin used herein were described previously (Charng et al., 2006; Chi et al., 2009).
2.3.8 Protein sequence alignment
Protein sequences of different species were obtained from NCBI
(http://www.ncbi.nlm.nih.gov). The alignment of the amino acid sequences of HSFs from different species was done by using the constraint-based multiple alignment tool (COBALT, http://www.ncbi.nlm.nih.gov/tools/cobalt) in NCBI.
2.4 Results
2.4.1 Isolation of triple and quadruple knockout lines of Arabidopsis HSFA1 genes To test the hypothesis that HSFA1 genes are functionally redundant in Arabidopsis, we produced a QK mutant of the Arabidopsis four HSFA1 genes by crossing the hsfA1a/1b (Ws ecotype background) and hsfA1d/1e (Col-0 ecotype background) mutants. In addition, to elucidate the function of each HSFA1 subtype, we isolated four triple KO mutants, named as aTK, bTK, dTK and eTK, where the prefixed letters represent the remaining functional HSFA1.
The genotypes and HSFA1 transcripts of these mutants were confirmed by PCR and RT-PCR, respectively (Fig. 2.1). Since the Col-0 and Ws genomic backgrounds were employed for generating the mutants, we included the wt of both ecotypes as controls.
2.4.2 The QK mutant showed growth retardation and altered morphologies
The Arabidopsis HSFA1 genes are not essential for viability and reproduction because the QK mutant could generate viable seeds at normal condition for several generations during our studies. However, in the absence of functional HSFA1 genes, plant growth and development were substantially affected. First, we noticed that the size of the mature seeds were
significantly decreased in the QK mutant but not in the triple KO mutants (Fig. 2.2a). We then
significantly decreased in the QK mutant but not in the triple KO mutants (Fig. 2.2a). We then