11.5: Recent Research on Motivation Theories
- Describe the modern advancements in the study of human motivation.
Employee motivation continues to be a major focus in organizational behavior (Ambrose & Kulik, 1999). We briefly summarize current motivation research here.
Content Theories
There is some interest in testing content theories (including Herzberg’s two-factor theory), especially in international research. Need theories are still generally supported, with most people identifying such workplace factors as recognition, advancement, and opportunities to learn as the chief motivators for them. This is consistent with need satisfaction theories. However, most of this research does not include actual measures of employee performance. Thus, questions remain about whether the factors that employees say motivate them to perform actually do.
Operant Conditioning Theory
There is considerable interest in operant conditioning theory, especially within the context of what has been called organizational behavior modification. Oddly enough, there has not been much research using operant conditioning theory in designing reward systems, even though there are obvious applications. Instead, much of the recent research on operant conditioning focuses on punishment and extinction. These studies seek to determine how to use punishment appropriately. Recent results still confirm that punishment should be used sparingly, should be used only after extinction does not work, and should not be excessive or destructive.
Equity Theory
Equity theory continues to receive strong research support. The major criticism of equity theory, that the inputs and outcomes people use to evaluate equity are ill-defined, still holds. Because each person defines inputs and outcomes, researchers are not in a position to know them all. Nevertheless, for the major inputs (performance) and outcomes (pay), the theory is a strong one. Major applications of equity theory in recent years incorporate and extend the theory into the area called organizational justice. When employees receive rewards (or punishments), they evaluate them in terms of their fairness (as discussed earlier). This is distributive justice. Employees also assess rewards in terms of how fair the processes used to distribute them are. This is procedural justice. Thus, during organizational downsizing, when employees lose their jobs, people ask whether the loss of work is fair (distributive justice). But they also assess the fairness of the process used to decide who is laid off (procedural justice). For example, layoffs based on seniority may be perceived as fairer than layoffs based on supervisors’ opinions.
Goal Theory
It remains true that difficult, specific goals result in better performance than easy and vague goals, assuming they are accepted. Recent research highlights the positive effects of performance feedback and goal commitment in the goal-setting process. Monetary incentives enhance motivation when they are tied to goal achievement, by increasing the level of goal commitment. There are negative sides to goal theory as well. If goals conflict, employees may sacrifice performance on important job duties. For example, if both quantitative and qualitative goals are set for performance, employees may emphasize quantity because this goal achievement is more visible.
Expectancy Theory
The original formulation of expectancy theory specifies that the motivational force for choosing a level of effort is a function of the multiplication of expectancies and valences. Recent research demonstrates that the individual components predict performance just as well, without being multiplied. This does not diminish the value of expectancy theory. Recent research also suggests that high performance results not only when the valence is high but also when employees set difficult goals for themselves.
One last comment on motivation: As the world of work changes, so will the methods organizations use to motivate employees. New rewards—time off instead of bonuses; stock options; on-site gyms, cleaners, and dental services; opportunities to telecommute; and others—will need to be created to motivate employees in the future. One useful path that modern researchers can undertake is to analyze the previous studies and aggregate the findings into more conclusive understanding of the topic through meta-analysis studies (Iddekinge et al., 2018).
CATCHING THE ENTREPRENEURIAL SPIRIT
Entrepreneurs and Motivation
Motivation can be difficult to elicit in employees. So what drives entrepreneurs, who by definition have to motivate themselves as well as others? While everyone from Greek philosophers to football coaches warn about undirected passion, a lack of passion will likely kill any start-up. An argument could be made that motivation is simply part of the discipline, or the outcome of remaining fixed on a purpose to mentally remind yourself of why you get up in the morning.
Working from her home in Egypt, at age 30 Yasmine El-Mehairy launched Supermama.me, a start-up aimed at providing information to mothers throughout the Arab world. When the company began, El-Mehairy worked full time at her day job and 60 hours a week after that getting the site established. She left her full-time job to manage the site full time in January 2011, and the site went live that October. El-Mehairy is motivated to keep moving forward, saying that if she stops, she might not get going again (Knowledge@Wharton, 2012).
For El-Mehairy, the motivation didn’t come from a desire to work for a big company or travel the world and secure a master’s degree from abroad. She had already done that. Rather, she said she was motivated to “do something that is useful and I want to do something on my own” (Knowledge@Wharton, 2012).
Lauren Lipcon, who founded a company called Injury Funds Now, attributes her ability to stay motivated to three factors: purpose, giving back, and having fun outside of work. Lipcon believes that most entrepreneurs are not motivated by money but by a sense of purpose. Personally, she left a job with Arthur Andersen to begin her own firm out of a desire to help people. She also thinks it is important for people to give back to their communities because the change the entrepreneur sees in the community loops back, increasing motivation and making the business more successful. Lipcon believes that having a life outside of work helps keep the entrepreneur motivated. She particularly advocates for physical activity, which not only helps the body physically but also helps keep the mind sharp and able to focus (Rashid, 2017).
But do all entrepreneurs agree on what motivates them? A July 17, 2017 survey on the entrepreneur's blog site asked 23 different entrepreneurs what motivated them. Seven of the 23 referred to some sense of purpose in what they were doing as a motivating factor, with one response stressing the importance of discovering one’s “personal why.” Of the remaining entrepreneurs, answers varied from keeping a positive attitude (three responses) and finding external sources (three responses) to meditation and prayer (two responses). One entrepreneur said his greatest motivator was fear: the fear of being in the same place financially one year in the future “causes me to take action and also alleviates my fear of risk” (Hear from Entrepreneurs, 2017). Only one of the 23 actually cited money and material success as a motivating factor to keep working.
However they describe motivation, entrepreneurs seem to agree that passion and determination are key factors that carry them through the grind of the day-to-day.
- In the article from Hear from Entrepreneurs, one respondent called motivation “garbage." Would you agree or disagree, and why?
- How is staying motivated as an entrepreneur similar to being motivated to pursue a college degree? Do you think the two are related? How?
- How would you expect motivation to vary across cultures?
- Understand the modern approaches to motivation theory.


7.6: Salinas v. Texas 570 US____ (2013)
NOTICE: This opinion is subject to formal revision before publication in the preliminary print of the United States Reports. Readers are requested to notify the Reporter of Decisions, Supreme Court of the United States, Washington, D. C. 20543, of any typographical or other formal errors, in order that corrections may be made before the preliminary print goes to press.
SUPREME COURT OF THE UNITED STATES
_________________
No. 12–246
_________________
GENOVEVO SALINAS, PETITIONER v. TEXAS
on writ of certiorari to the court of criminal appeals of texas
[June 17, 2013]
Justice Alito announced the judgment of the Court and delivered an opinion in which The Chief Justice and Justice Kennedy join.
Without being placed in custody or receiving Miranda warnings, petitioner voluntarily answered the questions of a police officer who was investigating a murder. But petitioner balked when the officer asked whether a ballistics test would show that the shell casings found at the crime scene would match petitioner’s shotgun. Petitioner was subsequently charged with murder, and at trial prosecutors argued that his reaction to the officer’s question suggested that he was guilty. Petitioner claims that this argument violated the Fifth Amendment, which guarantees that “[n]o person . . . shall be compelled in any criminal case to be a witness against himself.”
Petitioner’s Fifth Amendment claim fails because he did not expressly invoke the privilege against self-incrimination in response to the officer’s question. It has long been settled that the privilege “generally is not self-executing” and that a witness who desires its protection “ ‘must claim it.’ ” Minnesota v. Murphy, 465 U. S. 420, 425, 427 (1984) (quoting United States v. Monia, 317 U. S. 424, 427 (1943) ). Although “no ritualistic formula is necessary in order to invoke the privilege,” Quinn v. United States, 349 U. S. 155, 164 (1955) , a witness does not do so by simply standing mute. Because petitioner was required to assert the privilege in order to benefit from it, the judgment of the Texas Court of Criminal Appeals rejecting petitioner’s Fifth Amendment claim is affirmed.
I
On the morning of December 18, 1992, two brothers were shot and killed in their Houston home. There were no witnesses to the murders, but a neighbor who heard gunshots saw someone run out of the house and speed away in a dark-colored car. Police recovered six shotgun shell casings at the scene. The investigation led police to petitioner, who had been a guest at a party the victims hosted the night before they were killed. Police visited petitioner at his home, where they saw a dark blue car in the driveway. He agreed to hand over his shotgun for ballistics testing and to accompany police to the station for questioning.
Petitioner’s interview with the police lasted approximately one hour. All agree that the interview was noncustodial, and the parties litigated this case on the assumption that he was not read Miranda warnings. See Mi- randa v. Arizona, 384 U. S. 436 (1966) . For most of the interview, petitioner answered the officer’s questions. But when asked whether his shotgun “would match the shells recovered at the scene of the murder,” App. 17, petitioner declined to answer. Instead, petitioner “[l]ooked down at the floor, shuffled his feet, bit his bottom lip, cl[e]nched his hands in his lap, [and] began to tighten up.” Id., at 18. After a few moments of silence, the officer asked additional questions, which petitioner answered. Ibid.
Following the interview, police arrested petitioner on outstanding traffic warrants. Prosecutors soon concluded that there was insufficient evidence to charge him with the murders, and he was released. A few days later, police obtained a statement from a man who said he had heard petitioner confess to the killings. On the strength of that additional evidence, prosecutors decided to charge peti- tioner, but by this time he had absconded. In 2007, police discovered petitioner living in the Houston area under an assumed name.
Petitioner did not testify at trial. Over his objection, prosecutors used his reaction to the officer’s question dur- ing the 1993 interview as evidence of his guilt. The jury found petitioner guilty, and he received a 20-year sentence. On direct appeal to the Court of Appeals of Texas, petitioner argued that prosecutors’ use of his silence as part of their case in chief violated the Fifth Amendment. The Court of Appeals rejected that argument, reasoning that petitioner’s prearrest, pre-Miranda silence was not “compelled” within the meaning of the Fifth Amendment. 368 S. W. 3d 550, 557–559 (2011). The Texas Court of Criminal Appeals took up this case and affirmed on the same ground. 369 S. W. 3d 176 (2012).
We granted certiorari, 568 U. S. ___ (2013), to resolve a division of authority in the lower courts over whether the prosecution may use a defendant’s assertion of the privilege against self-incrimination during a noncustodial police interview as part of its case in chief. Compare, e.g., United States v. Rivera, 944 F. 2d 1563, 1568 (CA11 1991), with United States v. Moore, 104 F. 3d 377, 386 (CADC 1997). But because petitioner did not invoke the privilege during his interview, we find it unnecessary to reach that question.
II
A
The privilege against self-incrimination “is an exception to the general principle that the Government has the right to everyone’s testimony.” Garner v. United States, 424 U. S. 648 , n. 11 (1976). To prevent the privilege from shielding information not properly within its scope, we have long held that a witness who “ ‘desires the protection of the privilege . . . must claim it’ ” at the time he relies on it. Murphy, 465 U. S., at 427 (quoting Monia, 317 U. S., at 427). See also United States ex rel. Vajtauer v. Commissioner of Immigration, 273 U. S. 103, 113 (1927) .
That requirement ensures that the Government is put on notice when a witness intends to rely on the privilege so that it may either argue that the testimony sought could not be self-incriminating, see Hoffman v. United States, 341 U. S. 479, 486 (1951) , or cure any potential self-incrimination through a grant of immunity, see Kastigar v. United States, 406 U. S. 441, 448 (1972) . The express invocation requirement also gives courts tasked with evaluating a Fifth Amendment claim a contemporaneous record establishing the witness’ reasons for refusing to answer. See Roberts v. United States, 445 U. S. 552 , n. 7 (1980) (“A witness may not employ the privilege to avoid giving testimony that he simply would prefer not to give”); Hutcheson v. United States, 369 U. S. 599 –611 (1962) (declining to treat invocation of due process as proper assertion of the privilege). In these ways, insisting that witnesses expressly invoke the privilege “assures that the Government obtains all the information to which it is entitled.” Garner, supra, at 658, n. 11.
We have previously recognized two exceptions to the requirement that witnesses invoke the privilege, but neither applies here. First, we held in Griffin v. California, 380 U. S. 609 –615 (1965), that a criminal defendant need not take the stand and assert the privilege at his own trial. That exception reflects the fact that a criminal defendant has an “absolute right not to testify.” Turner v. United States, 396 U. S. 398, 433 (1970) (Black, J., dissenting); see United States v. Patane, 542 U. S. 630, 637 (2004) (plurality opinion). Since a defendant’s reasons for remaining silent at trial are irrelevant to his constitutional right to do so, requiring that he expressly invoke the privilege would serve no purpose; neither a showing that his testimony would not be self-incriminating nor a grant of immunity could force him to speak. Because pe- titioner had no comparable unqualified right during his interview with police, his silence falls outside the Griffin exception.
Second, we have held that a witness’ failure to invoke the privilege must be excused where governmental coercion makes his forfeiture of the privilege involuntary. Thus, in Miranda, we said that a suspect who is subjected to the “inherently compelling pressures” of an unwarned custodial interrogation need not invoke the privilege. 384 U. S., at 467–468, and n. 37. Due to the uniquely coercive nature of custodial interrogation, a suspect in custody cannot be said to have voluntarily forgone the privilege “unless [he] fails to claim [it] after being suitably warned.” Murphy, supra, at 429–430.
For similar reasons, we have held that threats to withdraw a governmental benefit such as public employment sometimes make exercise of the privilege so costly that it need not be affirmatively asserted. Garrity v. New Jersey, 385 U. S. 493, 497 (1967) (public employment). See also Lefkowitz v. Cunningham, 431 U. S. 801 –804 (1977) (public office); Lefkowitz v. Turley, 414 U. S. 70 –85 (1973) (public contracts). And where assertion of the privilege would itself tend to incriminate, we have allowed witnesses to exercise the privilege through silence. See, e.g., Leary v. United States, 395 U. S. 6 –29 (1969) (no requirement that taxpayer complete tax form where doing so would have revealed income from illegal activities); Albertson v. Subversive Activities Control Bd., 382 U. S. 70 –79 (1965) (members of the Communist Party not required to complete registration form “where response to any of the form’s questions . . . might involve [them] in the admission of a crucial element of a crime”). The principle that unites all of those cases is that a witness need not expressly invoke the privilege where some form of official compulsion denies him “a ‘free choice to admit, to deny, or to refuse to answer.’ ” Garner, 424 U. S., at 656–657 (quoting Lisenba v. California, 314 U. S. 219, 241 (1941) ).
Petitioner cannot benefit from that principle because it is undisputed that his interview with police was voluntary. As petitioner himself acknowledges, he agreed to accompany the officers to the station and “was free to leave at any time during the interview.” Brief for Petitioner 2–3 (internal quotation marks omitted). That places petitioner’s situation outside the scope of Miranda and other cases in which we have held that various forms of governmental coercion prevented defendants from voluntarily invoking the privilege. The dissent elides this point when it cites our precedents in this area for the proposition that “[c]ircumstances, rather than explicit invocation, trigger the protection of the Fifth Amendment.” Post, at 7–8 (opinion of Breyer, J.). The critical question is whether, under the “circumstances” of this case, petitioner was deprived of the ability to voluntarily invoke the Fifth Amendment. He was not. We have before us no allegation that petitioner’s failure to assert the privilege was involuntary, and it would have been a simple matter for him to say that he was not answering the officer’s question on Fifth Amendment grounds. Because he failed to do so, the prosecution’s use of his noncustodial silence did not violate the Fifth Amendment.
B
Petitioner urges us to adopt a third exception to the in- vocation requirement for cases in which a witness stands mute and thereby declines to give an answer that of- ficials suspect would be incriminating. Our cases all but foreclose such an exception, which would needlessly burden the Government’s interests in obtaining testimony and prosecuting criminal activity. We therefore decline petitioner’s invitation to craft a new exception to the “general rule” that a witness must assert the privilege to subsequently benefit from it. Murphy, 465 U. S., at 429.
Our cases establish that a defendant normally does not invoke the privilege by remaining silent. In Roberts v. United States, 445 U. S. 552 , for example, we rejected the Fifth Amendment claim of a defendant who remained silent throughout a police investigation and received a harsher sentence for his failure to cooperate. In so ruling, we explained that “if [the defendant] believed that his failure to cooperate was privileged, he should have said so at a time when the sentencing court could have determined whether his claim was legitimate.” Id., at 560. See also United States v. Sullivan, 274 U. S. 259 –264 (1927); Vajtauer, 273 U. S., at 113. [ 1 ] A witness does not expressly invoke the privilege by standing mute.
We have also repeatedly held that the express invocation requirement applies even when an official has reason to suspect that the answer to his question would incrim- inate the witness. Thus, in Murphy we held that the defendant’s self-incriminating answers to his probation of- ficer were properly admitted at trial because he failed to invoke the privilege. 465 U. S., at 427–428. In reaching that conclusion, we rejected the notion “that a witness must ‘put the Government on notice by formally availing himself of the privilege’ only when he alone ‘is reasonably aware of the incriminating tendency of the questions.’ ” Id., at 428 (quoting Roberts, supra, at 562, n.* (Brennan, J., concurring)). See also United States v. Kordel, 397 U. S. 1, 7 (1970) . [ 2 ]
Petitioner does not dispute the vitality of either of those lines of precedent but instead argues that we should adopt an exception for cases at their intersection. Thus, petitioner would have us hold that although neither a wit- ness’ silence nor official suspicions are enough to excuse the express invocation requirement, the invocation requirement does not apply where a witness is silent in the face of official suspicions. For the same reasons that neither of those factors is sufficient by itself to relieve a witness of the obligation to expressly invoke the privilege, we conclude that they do not do so together. A contrary result would do little to protect those genuinely relying on the Fifth Amendment privilege while placing a needless new burden on society’s interest in the admission of evidence that is probative of a criminal defendant’s guilt.
Petitioner’s proposed exception would also be very difficult to reconcile with Berghuis v. Thompkins, 560 U. S. 370 (2010) . There, we held in the closely related context of post-Miranda silence that a defendant failed to invoke the privilege when he refused to respond to police questioning for 2 hours and 45 minutes. 560 U. S., at ___ (slip op., at 3, 8–10). If the extended custodial silence in that case did not invoke the privilege, then surely the momentary silence in this case did not do so either.
Petitioner and the dissent attempt to distinguish Berg- huis by observing that it did not concern the admissi- bility of the defendant’s silence but instead involved the admissibility of his subsequent statements. Post, at 8–9 (opinion of Breyer, J.). But regardless of whether prosecutors seek to use silence or a confession that follows, the logic of Berghuis applies with equal force: A suspect who stands mute has not done enough to put police on notice that he is relying on his Fifth Amendment privilege. [ 3 ]
In support of their proposed exception to the invocation requirement, petitioner and the dissent argue that reliance on the Fifth Amendment privilege is the most likely explanation for silence in a case such as this one. Reply Brief 17; see post, at 9–10 (Breyer, J., dissenting). But whatever the most probable explanation, such silence is “insolubly ambiguous.” See Doyle, v. Ohio, 426 U. S. 610, 617 (1976) . To be sure, someone might decline to answer a police officer’s question in reliance on his constitutional privilege. But he also might do so because he is trying to think of a good lie, because he is embarrassed, or because he is protecting someone else. Not every such possible explanation for silence is probative of guilt, but neither is every possible explanation protected by the Fifth Amendment. Petitioner alone knew why he did not answer the officer’s question, and it was therefore his “burden . . . to make a timely assertion of the privilege.” Garner, 424 U. S., at 655.
At oral argument, counsel for petitioner suggested that it would be unfair to require a suspect unschooled in the particulars of legal doctrine to do anything more than remain silent in order to invoke his “right to remain silent.” Tr. of Oral Arg. 26–27; see post, at 10 (Breyer, J., dissenting); Michigan v. Tucker, 417 U. S. 433, 439 (1974) (observing that “virtually every schoolboy is familiar with the concept, if not the language” of the Fifth Amendment). But popular misconceptions notwithstanding, the Fifth Amendment guarantees that no one may be “compelled in any criminal case to be a witness against himself”; it does not establish an unqualified “right to remain silent.” A witness’ constitutional right to refuse to answer questions depends on his reasons for doing so, and courts need to know those reasons to evaluate the merits of a Fifth Amendment claim. See Hoffman, 341 U. S., at 486–487. [ 4 ]
In any event, it is settled that forfeiture of the privilege against self-incrimination need not be knowing. Murphy, 465 U. S., at 427–428; Garner, supra, at 654, n. 9. Statements against interest are regularly admitted into evidence at criminal trials, see Fed. Rule of Evid. 804(b)(3), and there is no good reason to approach a defendant’s silence any differently.
C
Finally, we are not persuaded by petitioner’s arguments that applying the usual express invocation requirement where a witness is silent during a noncustodial police interview will prove unworkable in practice. Petitioner and the dissent suggest that our approach will “unleash complicated and persistent litigation” over what a suspect must say to invoke the privilege, Reply Brief 18; see post, at 11–12 (opinion of Breyer, J.), but our cases have long required that a witness assert the privilege to subsequently benefit from it. That rule has not proved difficult to apply. Nor did the potential for close cases dissuade us from adopting similar invocation requirements for suspects who wish to assert their rights and cut off police questioning during custodial interviews. Berghuis, 560 U. S., at ___ (slip op., at 8–10) (requiring suspect to unambiguously assert privilege against self-incrimination to cut off custodial questioning); Davis v. United States, 512 U. S. 452, 459 (1994) (same standard for assertions of the right to counsel).
Notably, petitioner’s approach would produce its own line-drawing problems, as this case vividly illustrates. When the interviewing officer asked petitioner if his shotgun would match the shell casings found at the crime scene, petitioner did not merely remain silent; he made movements that suggested surprise and anxiety. At precisely what point such reactions transform “silence” into expressive conduct would be a difficult and recurring question that our decision allows us to avoid.
We also reject petitioner’s argument that an express invocation requirement will encourage police officers to “ ‘unfairly “tric[k]” ’ ” suspects into cooperating. Reply Brief 21 (quoting South Dakota v. Neville, 459 U. S. 553, 566 (1983) ). Petitioner worries that officers could unduly pressure suspects into talking by telling them that their silence could be used in a future prosecution. But as petitioner himself concedes, police officers “have done nothing wrong” when they “accurately stat[e] the law.” Brief for Petitioner 32. We found no constitutional infirmity in government officials telling the defendant in Murphy that he was required to speak truthfully to his parole officer, 465 U. S., at 436–438, and we see no greater danger in the interview tactics petitioner identifies. So long as police do not deprive a witness of the ability to voluntarily invoke the privilege, there is no Fifth Amendment violation.
* * *
Before petitioner could rely on the privilege against self-incrimination, he was required to invoke it. Because he failed to do so, the judgment of the Texas Court of Criminal Appeals is affirmed.
It is so ordered.
Notes
Contributors and Attributions
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In the business world, good ideas must pass one test: can they generate a profit? No idea is good, ultimately, if it can't pass this test. This means that someone, somewhere has to pay you more money for your product or service than it costs to produce it. Really, it's that simple-and that challenging. Perhaps the most important part of any business plan is an articulation of the customer. Why will they buy it? When? Where? For how much?
Of course other characteristics are important too: does the idea have a social good or at least do no harm? How challenging is it to make? Is it within the capabilities of the team? Sometimes student teams have great ideas but no way in the world of executing them. You may want to save your great idea until you've had a little more experience with something that's more manageable. Can you raise enough money to finance your idea? Often great ideas fail this test: simply put, they require too much capital to prove themselves.
You're not the only genius out there
So you've got a great, and maybe realistic, idea. You know what? Somebody else is probably working on a solution to the same problem. That's not necessarily a bad thing. A problem is only worth solving if it's a real problem that affects a real population. And you're probably not the only one who's noticed this problem. If you are, your product might be ahead of its time and have a hard time gaining acceptance in the marketplace. The best ideas don't happen in isolation; they're the result of a shared need. The need is what creates the market for your idea. The fact that others may have worked on addressing the same need heightens the importance of finding your own unique slant on the idea.
Talk about it
Talk about your idea with a lot of people, especially people with varied perspectives. But talk with people you trust. (And if you have any concerns about protecting your idea, talk to an intellectual property attorney-see the section "Your Intellectual Property.")  Talk with people with very different backgrounds from yours. Investigate companies that are doing work similar to yours, and visit some who are doing things that are totally different, and doing them well. See what you can learn from them. Listen to everyone responding to your idea. You may not agree with them, and you definitely don't have to take their advice, but there may be a gem hidden in even the most mundane or unpleasant piece of feedback.
Try it out
Experiment with your ideas while they're still in the raw stage.  Depending on the nature of your enterprise, your experimentation can take the form of actually making the product over and over until it comes out the way you want it, or running your ideas by a test audience to gauge its reaction.
Equip yourself
Take advantage of all the tools and resources available to you. Spend time in your school's library or entrepreneurial research center. Read books and magazines, watch videos (visit eclips.cornell.edu for a wealth of entrepreneurship-related video clips), and find out what experienced entrepreneurs have to say. Subscribe to an entrepreneurship magazine (Entrepreneur, Fast Company, Inc.), and spend time reading online articles. Find out what other resources they recommend, and if your school library doesn't have them, try inter-library loan. Join an entrepreneurship club or a similar community-based organization. Most importantly, get to know experienced entrepreneurs face-to-face, and listen to their stories.
Check in with yourself
Now why are you doing this again? Review your purpose in this project-what you hope to achieve, and what problem you hope to solve. It's okay if your purpose has changed since the project's outset. It will probably change many more times before you're through.
The validation phase
Once you've tested your idea with constituents, colleagues and mentors, and adjusted the idea to the point where you feel ready to take it to the public, seek external validation to get the funding and other support necessary to move ahead. Validation is the main purpose of a business plan. A business plan lays out your idea in a holistic manner so that prospective funders can enter into a partnership with confidence. See section 4, The Plan, for much more information on business plans.
|
Trial and error
Great ideas have three key elements
From "Hidden Asset," by Bill Breen, Fast Company, March 2004 www.fastcompany.com/magazine/80/davenport.html
A drive to fly
Research
Idea generation
Testing
Nonstop improvements
They moved their flight testing from Kitty Hawk to their home town of Dayton, Ohio, and flew their new aircraft at Huffman's Field on the edge of town. With new, more powerful aircraft, they were able to stay aloft for up to a half hour, fly figure eights, and even take passengers up for a ride. The age of the airplane had arrived. |


18.9C: Signs and Symptoms of Shock
The clinical manifestation of shock varies depending on the type of shock and the individual, but there are some general symptoms.
- Differentiate among the types of shock and the signs and symptoms associated with each
Key Points
- The general signs for all types of shock are low blood pressure, decreased urine output, and confusion. However, these may not always be present.
- Hypovolemic shock is characterized by loss of effective circulating blood volume, which leads to rapid pulse, cool skin, shallow breathing, hypothermia, thirst, and cold mottled skin.
- Cardiogenic shock is characterized by distended jugular veins, weak or absent pulse, and arrhythmia.
- Distributive shock includes septic shock, characterized by fever or anaphylaxis, and neurogenic shock, characterized by a reduced heart rate and vasodilation of superficial vessels warming the skin.
The presentation of shock varies. Some people presenting only minimal symptoms, such as confusion and weakness. Typical symptoms of shock include elevated but weak heart rate, low blood pressure, and poor organ function, typically observed as low urine output, confusion or loss of consciousness.
While a fast heart rate is common, those on beta blockers and those who are athletic may have a normal or slow heart rate. This also occurs in 30% of cases of shock caused by abdominal bleeding. Specific subtypes of shock may have additional symptoms.
Hypovolemic shock results from the direct loss of effective circulating blood volume. This leads to a rapid, weak pulse due to decreased blood flow combined with tachycardia, stimulation of vasoconstriction, and cool, clammy skin. It also presents with acidosis as well as rapid, shallow breathing due to sympathetic nervous system stimulation. Hypothermia due to decreased perfusion and evaporation of sweat, and thirst and dry mouth due to fluid depletion, may also be present.
Anaphalactic Hives : Hives and flushing on the back of a person with anaphylaxis.
The symptoms of cardiogenic shock are similar to those of hypovolemic shock. Additional symptoms may include arrhythmia of the heart beat and visibly distended jugular veins.
Distributive shock includes septic, anaphylactic, and neurogenic causes. With septic shock, fever may occur and the skin may be warm and sweaty. With anaphylaxis, hives may present on the skin, and there may localized edema, especially around the face, and weak and rapid pulse. Breathlessness and cough due to narrowed airways and swelling of the throat may also occur. The symptoms of neurogenic shock are distinct from those of classical shock, as the heart rate slows and and superficial vessels vasodilate and warm the skin. These symptoms are caused by neural damage and resultant loss of muscle control.
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17.2D: Enzymes Used in Industry
- Describe the use of enzymes in industry
Enzymes are biological molecules that catalyze (increase the rates of) chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates sufficient for life. Since enzymes are selective for their substrates and speed up only a few reactions from among many possibilities, the set of enzymes made in a cell determines which metabolic pathways occur in that cell.
Like all catalysts, enzymes work by lowering the activation energy for a reaction, thus dramatically increasing the rate of the reaction. As a result, products are formed faster and reactions reach their equilibrium state more rapidly. Most enzyme reaction rates are millions of times faster than those of comparable un-catalyzed reactions.
As with all catalysts, enzymes are not consumed by the reactions they catalyze, nor do they alter the equilibrium of these reactions. However, enzymes do differ from most other catalysts in that they are highly specific for their substrates. A few RNA molecules called ribozymes also catalyze reactions, with an important example being some parts of the ribosome. Synthetic molecules, called artificial enzymes, also display enzyme-like catalysis.
Enzyme activity can be affected by other molecules. Inhibitors can decrease enzyme activity; activators can increase activity. Many drugs and poisons are enzyme inhibitors. Activity is also affected by temperature, pressure, chemical environment (e.g., pH), and substrate concentration. Some enzymes are used commercially; for example, in the synthesis of antibiotics. In addition, some household products use enzymes to speed up biochemical reactions (e.g., enzymes in biological washing powders break down protein or fat stains on clothes; enzymes in meat tenderizers break down proteins into smaller molecules, making the meat easier to chew).
Enzymes are used in the chemical industry and other industrial applications when extremely specific catalysts are required. However, enzymes in general are limited in the number of reactions they have evolved to catalyze, and by their lack of stability in organic solvents and at high temperatures. As a consequence, protein engineering is an active area of research and involves attempts to create new enzymes with novel properties, either through rational design or in vitro evolution. These efforts have begun to be successful, and a few enzymes have now been designed “from scratch” to catalyze reactions that do not occur in nature.
In food processing, the enzymes used include amylases from fungi and plants. These enzymes are used in the production of sugars from starch, such as in making high-fructose corn syrup. In baking, they catalyze the breakdown of starch in the flour to sugar. Yeast fermentation of sugar produces the carbon dioxide that raises the dough. Proteases are used by biscuit manufacturers to lower the protein level of flour. Trypsin is used to predigest baby foods. For the processing of fruit juices, cellulases and pectinases are used to clarify fruit juices. Papain is used to tenderize meat for cooking.
In the dairy industry, rennin, derived from the stomachs of young ruminant animals (like calves and lambs) is used to manufacture of cheese, used to hydrolyze protein. Lipases are implemented during the production of Roquefort cheese to enhance the ripening of the blue-mold cheese. Lactases are used to break down lactose to glucose and galactose.
In the brewing industry, enzymes from barley are released during the mashing stage of beer production. They degrade starch and proteins to produce simple sugar, amino acids, and peptides that are used by yeast for fermentation. Industrially-produced barley enzymes are widely used in the brewing process to substitute for the natural enzymes found in barley. Amylase, glucanases, and proteases are used to split polysaccharides and proteins in the malt. Betaglucanases and arabinoxylanases are used to improve the wort and beer filtration characteristics. Amyloglucosidase and pullulanases are used for low-calorie beer and adjustment of fermentability. Proteases are used to remove cloudiness produced during storage of beers.
In the starch industry, amylases, amyloglucosideases, and glucoamylases convert starch into glucose and various syrups. Glucose isomerase converts glucose into fructose in production of high-fructose syrups from starchy materials.
In the paper industry, amylases, xylanases, cellulases, and ligninases are used to degrade starch to lower viscosity, aiding sizing and coating paper.
In the biofuel industry, cellulases used to break down cellulose into sugars that can be fermented (see cellulosic ethanol).
In the production of biological detergents, proteases, produced in an extracellular form from bacteria, are used in pre-soak conditions and direct liquid applications, helping with the removal of protein stains from clothes.
In molecular biology, restriction enzymes, DNA ligase, and polymerases are used to manipulate DNA in genetic engineering, important in pharmacology, agriculture and medicine, and are essential for restriction digestion and the polymerase chain reaction. Molecular biology is also important in forensic science.
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Key Points
- Since enzymes are selective for their substrates and speed up only a few reactions from among many possibilities, the set of enzymes made in a cell determines which metabolic pathways occur in that cell.
- Synthetic molecules called artificial enzymes also display enzyme-like catalysis.
- Enzymes are used in the chemical industry and other industrial applications when extremely specific catalysts are required.
Key Terms
- chemical reactions : Processes that lead to the transformation of one set of chemical substances to another.
- rennin : a proteolytic enzyme, obtained the gastric juice of the abomasum of calves, used to coagulate milk and make cheese
- enzymes : Biological molecules that catalyze (i.e., increase the rates of) chemical reactions.
- Synthetic : The combination of two or more parts, whether by design or by natural processes. It may imply being prepared or made artificially, in contrast to naturally.


9.4: Models for Population Growth
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Learning Objectives
In this section, we strive to understand the ideas generated by the following important questions:
- How can we use differential equations to realistically model the growth of a population?
- How can we assess the accuracy of our models?
The growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we will look at two ways in which we may use differential equations to help us address questions such as these. Before we begin, let’s consider again two important differential equations that we have seen in earlier work this chapter.
Preview Activity \(\PageIndex{1}\)
Recall that one model for population growth states that a population grows at a rate proportional to its size.
- We begin with the differential equation \[\dfrac{dP}{dt} = \dfrac{1}{2} P. \label{1}\] Sketch a slope field below as well as a few typical solutions on the axes provided.
- Find all equilibrium solutions of Equation \( \ref{1}\) and classify them as stable or unstable.
- If \(P(0)\) is positive, describe the long-term behavior of the solution to Equation \( \ref{1}\).
- Let’s now consider a modified differential equation given by \[\dfrac{dP}{dt} = \dfrac{1}{2} P(3 − P). \nonumber\] As before, sketch a slope field as well as a few typical solutions on the following axes provided.
- Find any equilibrium solutions and classify them as stable or unstable.
- If \(P(0)\) is positive, describe the long-term behavior of the solution.
The Earth’s Population
We will now begin studying the earth’s population. To get started, here are some data for the earth’s population in recent years that we will use in our investigations.
| Year | 1998 | 1999 | 2000 | 2001 | 2002 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Population (in Billions) | 5.932 | 6.008 | 6.084 | 6.159 | 6.234 | 6.456 | 6.531 | 6.606 | 6.681 | 6.756 | 6.831 |
Activity \(\PageIndex{1}\): Growth Dynamics
Our first model will be based on the following assumption:
The rate of change of the population is proportional to the population.
On the face of it, this seems pretty reasonable. When there is a relatively small number of people, there will be fewer births and deaths so the rate of change will be small. When there is a larger number of people, there will be more births and deaths so we expect a larger rate of change. If \(P(t)\) is the population \(t\) years after the year 2000, we may express this assumption as \[\dfrac{dP}{ dt} = kP \label{eq2}\]
where \(k\) is a constant of proportionality.
- Use the data in the table to estimate the derivative \(P'(0)\) using a central difference. Assume that \(t = 0\) corresponds to the year 2000.
- What is the population \(P(0)\)?
- Use these two facts to estimate the constant of proportionality \(k \)in the differential equation.
- Now that we know the value of \(k\), we have the initial value problem of Equation \( \ref{eq2}\) with \(P(0) = 6.084\). Find the solution to this initial value problem.
- What does your solution predict for the population in the year 2010? Is this close to the actual population given in the table?
- When does your solution predict that the population will reach 12 billion?
- What does your solution predict for the population in the year 2500?
- Do you think this is a reasonable model for the earth’s population? Why or why not? Explain your thinking using a couple of complete sentences.
Our work in Activity \(\PageIndex{1}\) shows that that the exponential model is fairly accurate for years relatively close to 2000. However, if we go too far into the future, the model predicts increasingly large rates of change, which causes the population to grow arbitrarily large. This does not make much sense since it is unrealistic to expect that the earth would be able to support such a large population.
The constant \(k\) in the differential equation has an important interpretation. Let’s rewrite the differential equation
\(\dfrac{dP}{dt} = kP\)
by solving for \(k\), so that we have
\(k = \dfrac{\dfrac{dP}{dt}}{ P}.\)
Viewed in this light, \(k\) is the ratio of the rate of change to the population; in other words, it is the contribution to the rate of change from a single person. We call this the per capita growth rate .
In the exponential model we introduced in Activity \(\PageIndex{1}\), the per capita growth rate is constant. In particular, we are assuming that when the population is large, the per capita growth rate is the same as when the population is small. It is natural to think that the per capita growth rate should decrease when the population becomes large, since there will not be enough resources to support so many people. In other words, we expect that a more realistic model would hold if we assume that the per capita growth rate depends on the population P. In the previous activity, we computed the per capita growth rate in a single year by computing \(k\), the quotient of \(\frac{dP}{dt}\) and \(P\) (which we did for \(t = 0\)). If we return data and compute the per capita growth rate over a range of years, we generate the data shown in Figure \(\PageIndex{1}\), which shows how the per capita growth rate is a function of the population, \(P\).
Figure \(\PageIndex{1}\): A plot of per capita growth rate vs. population P.
From the data, we see that the per capita growth rate appears to decrease as the population increases. In fact, the points seem to lie very close to a line, which is shown at two different scales in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\): The line that approximates per capita growth as a function of population, P.
Looking at this line carefully, we can find its equation to be
\(\dfrac{\dfrac{dP}{dt}}{ P} = 0.025 − 0.002P.\)
If we multiply both sides by \(P\), we arrive at the differential equation
\[\dfrac{dP}{ dt} = P(0.025 − 0.002P). \label{log}\]
Graphing the dependence of \(\frac{dP}{dt}\) on the population \(P\), we see that this differential equation demonstrates a quadratic relationship between \(\frac{dP}{dt}\) and \(P\), as shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): A plot of \(\frac{dP}{dt}\) vs. \(P\) for Equation \(\ref{log}\).
Equation \( \ref{log}\) is an example of the logistic equation , and is the second model for population growth that we will consider. We have reason to believe that it will be more realistic since the per capita growth rate is a decreasing function of the population.
Indeed, the graph in Figure \(\PageIndex{3}\) shows that there are two equilibrium solutions, \(P = 0\), which is unstable, and \(P = 12.5\), which is a stable equilibrium. The graph shows that any solution with \(P(0) > 0\) will eventually stabilize around 12.5. In other words, our model predicts the world’s population will eventually stabilize around 12.5 billion.
A prediction for the long-term behavior of the population is a valuable conclusion to draw from our differential equation. We would, however, like to answer some quantitative questions. For instance, how long will it take to reach a population of 10 billion? To determine this, we need to find an explicit solution of the equation. Solving the logistic differential equation Since we would like to apply the logistic model in more general situations, we state the logistic equation in its more general form,
\[\dfrac{dP}{ dt} = kP(N − P). \label{7.2} \]
The equilibrium solutions here are when \(P = 0\) and \(1 − \frac{P}{N} = 0\), which shows that \(P = N\). The equilibrium at \(P = N\) is called the carrying capacity of the population for it represents the stable population that can be sustained by the environment.
We now solve the logistic Equation \( \ref{7.2}\), which is separable, so we separate the variables
\(\dfrac{1}{P(N − P)} \dfrac{ dP}{ dt} = k, \)
and integrate to find that
\( \int \dfrac{1}{P(N − P)} dP = \int k dt, \)
To find the antiderivative on the left, we use the partial fraction decomposition
\(\dfrac{1}{P(N − P)} = \dfrac{1}{ N} \left[ \dfrac{ 1}{ P} + \dfrac{1}{ N − P} \right] .\)
Now we are ready to integrate, with
\( \int \dfrac{1}{ N} \left[ \dfrac{1}{ P} + \dfrac{1}{ N − P} \right] dP = \int k dt.\)
On the left, observe that \(N\) is constant, so we can remove the factor of \(\frac{1}{N}\) and antidifferentiate to find that
\(\dfrac{1}{ N} (\ln |P| − \ln |N − P|) = kt + C. \)
Multiplying both sides of this last equation by \(N\) and using an important rule of logarithms, we next find that
\( \ln \left| \dfrac{P}{ N − P} \right | = kNt + C. \)
From the definition of the logarithm, replacing \(e^C\) with \(C\), and letting \(C\) absorb the absolute value signs, we now know that
\( \dfrac{P}{N − P} = Ce^{k N t}.\)
At this point, all that remains is to determine \(C\) and solve algebraically for \(P\).
If the initial population is \(P(0) = P_0\), then it follows that
\(C = \dfrac{P_0}{ N−P_0}\)
so
\(\dfrac{P}{N − P} = \dfrac{P_0}{ N − P_0} e^{ k N t} .\)
We will solve this most recent equation for \(P\) by multiplying both sides by \((N − P)(N − P_0)\) to obtain
\( \begin{align} P(N − P_0) & = P_0(N − P)e^{k N t} \\ & = P_0Ne^{k N t} − P_0Pe^{k N t}. \end{align}\)
Swapping the left and right sides, expanding, and factoring, it follows that
\(\begin{align} P_0Ne^{k N t} & = P(N − P_0) + P_0Pe^{k N t} \\ & = P(N − P_0 + P_0e^{ k N t}). \end{align}\)
Dividing to solve for \(P\), we see that
\(P = \dfrac{P_0Ne^{k N t}}{ N − P_0 + P_0e^{k N t}}.\)
Finally, we choose to multiply the numerator and denominator by \(\frac{1}{P_0} e^{−k N t}\) to obtain
\[P(t) = \dfrac{N}{ \left( \dfrac{N−P_0}{P_0} \right) e^{−k N t} + 1} . \label{7.3}\]
While that was a lot of algebra, notice the result: we have found an explicit solution to the initial value problem
\(\dfrac{dP}{dt} = kP(N − P),\ P(0)=P_{0},\)
with \(P(0) = P_0\) and that solution is Equation \( \ref{7.3}\).
For the logistic equation describing the earth’s population that we worked with earlier in this section, we have
\(k = 0.002\), \(N = 12.5\), and \(P_0 = 6.084\).
This gives the solution
\[P(t) = \dfrac{12.5}{ 1.0546e^{−0.025t} + 1}, \label{earth}\]
whose graph is shown in Figure \(\PageIndex{4}\) Notice that the graph shows the population leveling off at 12.5 billion, as we expected, and that the population will be around 10 billion in the year 2050. These results, which we have found using a relatively simple mathematical model, agree fairly well with predictions made using a much more sophisticated model developed by the United Nations.
Figure \(\PageIndex{4}\): The solution to the logistic equation modeling the earth’s population (Equation \ref{earth}).
The logistic equation is useful in other situations, too, as it is good for modeling any situation in which limited growth is possible. For instance, it could model the spread of a flu virus through a population contained on a cruise ship, the rate at which a rumor spreads within a small town, or the behavior of an animal population on an island. Again, it is important to realize that through our work in this section, we have completely solved the logistic equation, regardless of the values of the constants \(N\), \(k\), and \(P_0\).
Anytime we encounter a logistic equation, we can apply the formula we found in Equation \ref{7.3}.
Activity \(\PageIndex{2}\): Predicting Earth's Population
Consider the logistic equation
\(\dfrac{dP}{dt} = kP(N − P) \nonumber\)
with the graph of \(\frac{dP}{dt}\) vs. \(P\) shown below.
- At what value of \(P\) is the rate of change greatest?
- Consider the model for the earth’s population that we created. At what value of \(P\) is the rate of change greatest? How does that compare to the population in recent years?
- According to the model we developed, what will the population be in the year 2100?
- According to the model we developed, when will the population reach 9 billion?
- Now consider the general solution to the general logistic initial value problem that we found, given by Equation \( \ref{7.3}\). Verify algebraically that \(P(0) = P_0\) and that \(\lim_{t→\infty} P(t) = N.\)
Summary
In this section, we encountered the following important ideas:
- If we assume that the rate of growth of a population is proportional to the population, we are led to a model in which the population grows without bound and at a rate that grows without bound.
- By assuming that the per capita growth rate decreases as the population grows, we are led to the logistic model of population growth, which predicts that the population will eventually stabilize at the carrying capacity.


1.3: Hypothesis, Theories, and Laws
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Learning Objectives
- Describe the difference between hypothesis and theory as scientific terms.
- Describe the difference between a theory and scientific law.
Although many have taken science classes throughout the course of their studies, people often have incorrect or misleading ideas about some of the most important and basic principles in science. Most students have heard of hypotheses, theories, and laws, but what do these terms really mean? Prior to reading this section, consider what you have learned about these terms before. What do these terms mean to you? What do you read that contradicts or supports what you thought?
What is a Fact?
A fact is a basic statement established by experiment or observation. All facts are true under the specific conditions of the observation.
What is a Hypothesis?
One of the most common terms used in science classes is a "hypothesis". The word can have many different definitions, depending on the context in which it is being used:
- An educated guess: a scientific hypothesis provides a suggested solution based on evidence.
- Prediction: if you have ever carried out a science experiment, you probably made this type of hypothesis when you predicted the outcome of your experiment.
- Tentative or proposed explanation: hypotheses can be suggestions about why something is observed. In order for it to be scientific, however, a scientist must be able to test the explanation to see if it works and if it is able to correctly predict what will happen in a situation. For example, "if my hypothesis is correct, we should see ___ result when we perform ___ test."
A hypothesis is very tentative; it can be easily changed.
What is a Theory?
The United States National Academy of Sciences describes what a theory is as follows:
"Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature supported by facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena."
"A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experimentation. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter (stating that everything is made of atoms) or the germ theory of disease (which states that many diseases are caused by germs). Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.
Note some key features of theories that are important to understand from this description:
- Theories are explanations of natural phenomena. They aren't predictions (although we may use theories to make predictions). They are explanations as to why we observe something.
- Theories aren't likely to change. They have a large amount of support and are able to satisfactorily explain numerous observations. Theories can, indeed, be facts. Theories can change, but it is a long and difficult process. In order for a theory to change, there must be many observations or pieces of evidence that the theory cannot explain.
- Theories are not guesses. The phrase "just a theory" has no room in science. To be a scientific theory carries a lot of weight; it is not just one person's idea about something
Theories aren't likely to change.
What is a Law?
Scientific laws are similar to scientific theories in that they are principles that can be used to predict the behavior of the natural world. Both scientific laws and scientific theories are typically well-supported by observations and/or experimental evidence. Usually scientific laws refer to rules for how nature will behave under certain conditions, frequently written as an equation. Scientific theories are more overarching explanations of how nature works and why it exhibits certain characteristics. As a comparison, theories explain why we observe what we do and laws describe what happens.
For example, around the year 1800, Jacques Charles and other scientists were working with gases to, among other reasons, improve the design of the hot air balloon. These scientists found, after many, many tests, that certain patterns existed in the observations on gas behavior. If the temperature of the gas is increased, the volume of the gas increased. This is known as a natural law. A law is a relationship that exists between variables in a group of data. Laws describe the patterns we see in large amounts of data, but do not describe why the patterns exist.
What is a Belief?
A belief is a statement that is not scientifically provable. Beliefs may or may not be incorrect; they just are outside the realm of science to explore.
Laws vs. Theories
A common misconception is that scientific theories are rudimentary ideas that will eventually graduate into scientific laws when enough data and evidence has accumulated. A theory does not change into a scientific law with the accumulation of new or better evidence. Remember, theories are explanations and laws are patterns we see in large amounts of data, frequently written as an equation. A theory will always remain a theory; a law will always remain a law.
Video \(\PageIndex{1}\): What’s the difference between a scientific law and theory?
Summary
- A hypothesis is a tentative explanation that can be tested by further investigation.
- A theory is a well-supported explanation of observations.
- A scientific law is a statement that summarizes the relationship between variables.
- An experiment is a controlled method of testing a hypothesis.


2.3: Divisibility in Integral Domains
In this section, we'll seek to answer the questions:
- What multiplicative properties can we generalize from \(\mathbb{Z}\) to any integral domain?
- What are the differences between a prime and irreducible element in a commutative ring?
When we introduced the notion of integral domain, we said that part of the reason for the definition was to capture some of the most essential properties of the integers. This is the heart of abstraction and generalization in mathematics: to distill the important properties of our objects of interest and explore the consequences of those properties. One such important property of \(\mathbb{Z}\) is cancellation .
Let \(R\) be a ring. Then \(R\) is a domain if and only if for all \(a,b,c\in R\) with \(c\ne 0\) and \(ac = bc\text{,}\) we have \(a = b\text{.}\)
We may read Theorem 2.3.1 as saying that the defining property of an integral domain is the ability to cancel common nonzero factors. Note that we have not divided ; division is not a binary operation, and nonzero elements of rings need not be units. However, as was the case in \(\mathbb{Z}\text{,}\) there are notions of divisibility and factorization in rings.
Let \(R\) be a commutative ring with identity, and let \(a,b\in R\text{.}\) We say \(a\) divides \(b\) and write \(a\mid b\) if there is a \(c\in R\) such that \(ac = b\text{.}\) We then say that \(a\) is a factor of \(b\text{.}\)
Find all factors of \(\overline{2}\) in the following rings:
- \(\displaystyle \mathbb{Z}_5\)
- \(\displaystyle \mathbb{Z}_6\)
- \(\displaystyle \mathbb{Z}_{10}\)
Our definition of prime also extends nicely to domains. Indeed, the desire to extend the familiar notion of prime from \(\mathbb{Z}\) to any ring is the reason for our less-familiar definition given in Definition: Prime .
Let \(R\) be a domain. We say a nonzero nonunit element \(a\in R\) is prime if whenever \(a\mid bc\) for some \(b,c\in R\text{,}\) either \(a|b\) or \(a|c\text{.}\)
A notion related to primality is irreducibility. In fact, one might reasonably say that irreducibility is the natural generalization of the typical definition of prime one encounters in school mathematics.
Let \(R\) be a domain. We say a nonzero nonunit element \(a\in R\) is irreducible if whenever \(a = bc\) for some \(b,c\in R\text{,}\) one of \(b\) or \(c\) is a unit. (Note that in some areas of the literature, the word atom is used interchangeably with irreducible.)
Find the units, primes, and irreducibles in the following rings.
- \(\displaystyle \mathbb{R}\)
- \(\displaystyle \mathbb{Z}\)
- \(\displaystyle \mathbb{Z}_{5}\)
- \(\displaystyle \mathbb{Z}_6\)
In domains, all primes are irreducible.
Let \(R\) be a domain. If \(a\in R\) is prime, then \(a\) is irreducible.
In familiar settings, the notion of prime and irreducible exactly coincide.
Every irreducible in \(\mathbb{Z}\) is prime.
Despite their overlap in familiar settings, primes and irreducibles are distinct types of elements. As the next exploration demonstrates, not all primes are irreducible. What is more, Exploration 2.3.3 will show that not all irreducibles are primes, even in domains!
Find an example of a ring \(R\) and prime \(p\in R\) such that \(p\) is not irreducible.
Consider the set \(R\) of all polynomials in \(\mathbb{Z}[x]\) for which the coefficient on the linear term is zero. That is,
\begin{equation*} R = \{a_0 + a_2 x^2 + \cdots + a_{n-1} x^{n-1} + a_n x^n: a_i\in \mathbb{Z},\ n\in\mathbb{N}_0 \}\text{.} \end{equation*}
(You should convince yourself that \(R\) is an integral domain, but do not need to prove it.) Then, find a polynomial of the form \(x^n\) in \(R\) that is irreducible, but not prime.
Our last straightforward generalization from the multiplicative structure of \(\mathbb{Z}\) is the notion of greatest common divisor. As our next definition again demonstrates, our careful work in the context of \(\mathbb{Z}\) generalizes nicely to all domains. Indeed, we intentionally did not appeal to \(\le\) to define the greatest common divisor in Definition: Greatest Common Divisor , as not all rings have a natural order relation like \(\mathbb{Z}\) does.
Let \(R\) be a domain, and let \(a,b\in R\text{.}\) A nonzero element \(d\in R\) is a greatest common divisor of \(a\) and \(b\) if
- \(d\mid a\) and \(d\mid b\) and,
- if \(e\in R\) with \(e\mid a\) and \(e\mid b\text{,}\) then \(e\mid d\text{.}\)
Let \(R\) be a domain and \(a,b\in R\) and suppose \(d\) is a greatest common divisor of \(a\) and \(b\text{.}\) Then any associate of \(d\) is also a greatest common divisor of \(a\) and \(b\text{.}\) (Recall Definition: Unit )
In most familiar domains, GCDs exist. However, they don't always! Find an example of elements in the ring from Exploration 2.3.3 which do not have a GCD. Justify your assertion.
Fill in the following blanks in order of increasing generality with the words ring , integral domain , field , and commutative ring .
__________ \(\Rightarrow\) __________ \(\Rightarrow\) __________ \(\Rightarrow\) __________


4.2: Headers and Footers
Learning Objectives
- How to create headers and footers for a document.
Header and Footer are to insert whatever you need (such as page numbers, title, date…Etc.) into the head (top) and the foot (bottom) of each page. However, the main purpose for the header and footer is to add various scripts to track the access to the blog (such as Google…Etc.).
To create a header or a footer:
- Click the Insert tab.
- Click on the Header/Footer pop-down menu.
After you insert the header/footer, type in header or footer whatever you like…
- You can insert page numbers as it shown in the picture above by choosing the design you like.
- Also, you can change the height of the Header/footer from the Position group as it shown in the picture above.
- After finishing editing the header/footer, to exit click the (" Close Header and Footer", the red X next to the Position group).


4.5: Gene Therapy
Gene therapy is a genetic engineering technique used to cure disease. In its simplest form, it involves the introduction of a good gene at a random location in the genome to aid the cure of a disease that is caused by a mutated gene. In theory, the introduction of a good gene will allow the organism to produce a functioning protein instead of the mutated or absent version caused by a disease. Gene therapy attempts have often involved introducing the good gene into diseased cells as part of a vector transmitted by a virus. The virus vector (containing the good gene) can then infect the host cell and deliver the foreign DNA to the diseased cells (Figure 2).
Figure 2: Gene therapy using an adenovirus vector can be used to cure certain genetic diseases in which a person has a defective gene. (credit: OpenStax Microbiology 4.4. CC-BY.)
CRISPR/Cas9
More recently, there has been much interest in correcting mutated genes using CRISPR/Cas9 technology that was discovered in bacteria. CRISPR stands for Clustered Regularly InterSpaced Palindromic Repeats sequences that exist in bacterial chromosomes. Bacteria insert foreign DNA into the CRISPR sequence and use that as a template or guide to survey the cell for that foreign DNA sequence and then cleave the foreign target DNA using the Cas9 enzyme. Thus, the CRISPR/Cas9 system is a form of bacterial immune system to recognize and cleave invading DNA from bacterial viruses.
CRISPR/Cas9 involves two essential components: a guide RNA to match a desired target gene, and Cas9 (CRISPR-associated protein 9), an endonuclease which causes a double-stranded DNA break, allowing modifications to the genome (see figure 3). Cas9 is the enzyme that cuts the target DNA. Scientists have discovered that CRISPR technology makes it possible to correct errors in the genome and turn on or off genes in cells and organisms quickly, cheaply and with relative ease. One of the most exciting applications of CRISPR/Cas9 is its potential use to treat genetic disorders caused by single gene mutations. Examples of such diseases include cystic fibrosis, Duchenne's muscular dystrophy (DMD) and haemoglobinopathies. Scientists have used CRISPR/Cas9 technology to restore cystic fibrosis transmembrane conductor receptor function in stem cells from cystic fibrosis patients. Researchers have also used CRISPR/Cas9 technology to restore muscle function by recovering dystrophin protein expression in a mouse model of Duchenne’s muscular dystrophy. The approach so far has currently only been validated in preclinical models, but there is hope it can soon be translated to clinical practice. Other potential clinical applications include gene therapy, treating infectious diseases such as HIV and engineering autologous patient material to treat cancer and other diseases. There has been increasing interest in the possibility of using CRISPR/Cas9 to modify patient-derived T-cells and stem/progenitor cells which can then be reintroduced into patients to treat disease, such as cancers.
Overview of CRISPR/Cas9
Figure 3
The CRISPR/Cas9 system. Clustered regularly interspaced palindromic repeats (CRISPR) refers to sequences in the bacterial genome. They afford protection against invading viruses, when combined with a series of CRISPR-associated (Cas) proteins. Cas9, one of the associated proteins, is an endonuclease that cuts both strands of DNA. Cas9 is directed to its target by a section of RNA. This can be synthesized as a single strand called a synthetic single guide RNA (sgRNA); the section of RNA which binds to the genomic DNA is 18–20 nucleotides. In order to cut, a specific sequence of DNA of between 2 and 5 nucleotides (the exact sequence depends upon the bacteria which produces the Cas9) must lie at the 3’ end of the guide RNA: this is called the protospacer adjacent motif (PAM). Repair after the DNA cut may occur via two pathways: non-homologous end joining, typically leading to a random insertion/deletion of DNA, or homology directed repair where a homologous piece of DNA is used as a repair template. It is the latter which allows precise genome editing: the homologous section of DNA with the required sequence change may be delivered with the Cas9 nuclease and sgRNA, theoretically allowing changes as precise as a single base-pair.
The therapeutic applications of CRISPR/Cas9 investigated have predominantly been directed at somatic cells. A particularly controversial issue surrounding CRISPR/Cas9 is that of gene editing in embryos. It has already been shown that CRISPR/Cas9 technology can alter the genome of human embryos3 which theoretically could prove useful in the preimplantation treatment of genetic diseases. However, any genetic modification of the germline would be permanent and the long-term consequences are unclear. Many oppose germline modification under any circumstances, reasoning that an eventual consequence could be non-therapeutic genetic enhancement.13It is clear that the ethical boundaries, within which CRISPR/Cas9 can be used, remain to be fully determined.
Microarrays
Some diseases, such as cancer, result from cells producing too much or too little of a protein. Microarrays are used to assess the levels of mRNA present in cells, which is indicative of the levels of proteins within cells. Microarray analysis is useful for the comparison of gene-expression patterns between different cell types—for example, cells infected with a virus versus uninfected cells, or cancerous cells versus healthy cells. Microarrays allow scientists to detect the expression of thousands of genes in one assay.
Microarray microscope slides, often called gene chips, are spotted with thousands of tiny spots containing different known DNA sequences. These DNA sequences act as probes. mRNA is isolated from cells (for example: mRNA could be isolated from a diseased cell and compared to mRNA from a healthy cell). The isolated mRNA is converted to a single stranded complementary DNA copy, cDNA, by using an enzyme called reverse transcriptase. The cDNA is labeled with a fluorescent dye and then allowed to bind to the microarray slide. cDNA will bind, or hybridize, to complementary DNA sequences on the microarray slide. Following time for hybridization to occur, the microarray slide is scanned for levels of fluorescence for each gene on the slide. Greater fluorescence associated with a gene indicates that the sample contained high levels of mRNA. No fluorescence for a specific gene indicates that the sample did not contain mRNA for that gene.
Often when performing microarrays, the cDNA from two different samples are labeled with two different fluorescent dyes (typically red and green). The fluorescently labeled cDNA samples are combined in equal amounts, added to the microarray chip, and allowed to hybridize to complementary spots on the microarray. Hybridization of sample genomic DNA molecules can be monitored by measuring the intensity of fluorescence at particular spots on the microarray. Differences in the amount of hybridization between the samples can be readily observed. If only one sample’s nucleic acids hybridize to a particular spot on the microarray, then that spot will appear either green or red. However, if both samples’ nucleic acids hybridize, then the spot will appear yellow due to the combination of the red and green dyes (Figure 4). When used in this way, microarrays can generate gene expression profiles that show differences in mRNA levels of many genes when comparing control/untreated cells to diseased or treated cells.
Figure 4: Microarray to compare mRNA levels. (a) The steps in microarray analysis are illustrated. Here, gene expression patterns are compared between cancerous cells and healthy cells. (b) Microarray information can be expressed as a heat map. Genes are shown on the left side; different samples are shown across the bottom. Genes expressed only in cancer cells are shown in varying shades of red; genes expressed only in normal cells are shown in varying shades of green. Genes that are expressed in both cancerous and normal cells are shown in yellow. Microarray CNX OpenStax, CC BY 4.0, via Wikimedia Commons.
Questions for Review
- When a gene chip is used in a microarray, what is spotted onto the microscope slide to create the gene chip?
- If someone is trying to determine if a cell expresses a protein that is known to be expressed in cancerous cells, what molecule would be isolated from the cells to be used in a microarray?
- Reverse transcriptase is mixed with the isolated samples to generate what molecule?
- What allows us to visualize if the samples bind to the microarray?
- What color shows on the microarray if both samples have bound to a sequence on the array?
mRNA Vaccines
In the 1970’s there was much interest in introducing mRNA into cells as a method of gene therapy. In theory, the mRNA would be translated directly within the cells to produce a correctly folded and functional gene, thus treating the disease. However, mRNA produced in vitro was found to elicit a strong inflammatory immune response and did not produce good or long lasting gene expression.
mRNA molecules are infamously known to be degraded quickly in the lab and in the body. Scientists who studied mRNA molecules for use in gene therapy had difficulty delivering the mRNA to the target cells before it would get quickly degraded by the body. In the late 1980’s scientists were beginning to uncover that enveloping mRNA in lipids (fats), creating a fatty nanoparticle, allows the mRNA to be successfully delivered to target cells without being quickly degraded before the mRNA can be translated to protein.
In 1998, Dr. Katalin Karikó, who worked extensively on mRNA, and Dr. Drew Weissman, an immunologist, began to collaborate to investigate the potential use of mRNA to generate vaccines. In theory, they hoped mRNA could be safely delivered to cells, then the mRNA is translated in the cytoplasm of the cells to generate a specific protein (generally a protein that is part of or produced by a specific pathogenic bacteria or virus). The body then mounts an immune response against the bacterial/viral protein and that memory immune response protects the person against that specific pathogen if it is encountered in the future. Initially, when mRNA vaccines were injected into mice they suffered severe inflammatory responses and some of the mice died. But, in 2005, Drs. Karikó and Weissman, published results showing that altering one of the mRNA building blocks, nucleosides, allowed mRNA to be less inflammatory and safely used therapeutically. It was becoming clear to their scientific team that it was possible to practically and safely use mRNA therapeutically. Early studies investigated the use of mRNA vaccines to prevent Ebola virus, but there was limited funding and demand for mRNA Ebola vaccines.
The COVID-19 pandemic brought a tremendous influx of funding and motivation to further investigate the use of mRNA vaccines to prevent or treat illness. Several different teams worked to develop mRNA COVID-19 vaccines. Extensive clinical trials showed that COVID-19 mRNA vaccines are both safe and effective. As of 2022, two independently developed mRNA COVID-19 vaccines have received full FDA approval, these are the first mRNA products to receive full FDA approval. mRNA has proven to be a powerful life-saving tool in the generation of vaccines. mRNA can be relatively easily and quickly manipulated, which makes mRNA vaccines the ideal method for the generation of vaccines against new and emerging deadly infectious diseases.
- Sometimes children who receive the chickenpox vaccine will develop minor chickenpox blisters/rash at the sight of the injection because this vaccine contains a weakened form of the chickenpox virus (Varicella zoster virus). Is it possible for someone to contract COVID-19 infection as a result of receiving a COVID-19 mRNA vaccine? Explain why or why not.
- Some people are concerned about mRNA vaccines modifying human DNA within the nucleus. Does this happen? Explain why or why not.
- When the COVID-19 mRNA vaccine enters our cells it is translated to make the viral spike protein. Does this mean that our cells will make the spike protein for the rest of our lives?
- How are mRNA vaccines considered to be safer than some other types of vaccines?
Questions for Review
- Detergent is added to DNA extraction buffer to ____________.
- Why does DNA precipitate when alcohol is added to the DNA extraction buffer?
- In genetic engineering, scientists make a specific change in the ___________ of an organism.
- The CRISPR/Cas9 system originated from what organisms? What purpose does this system serve in these organisms?
- Microarrays detect differences in levels of __________ in organisms.
Testing Understanding Post-Lab
- Sickle cell anemia is a genetic disease caused by a mutation in a single gene. This mutation causes the hemoglobin protein to be misfolded and therefore not able to carry as much oxygen throughout the body. Explain how CRISPR/Cas9 could possibly be used therapeutically to treat this genetic disease?